Sylvain Lenfle
email: sylvain.lenfle@lecnam.net

-1960)

The current skepticism about basic research in industry imagines a world of detached corportate laboratory that existed only for a moment in a much longer history of successful balancing of the commitment to the long term with the need for a commercial payoff. This historical misperception perpetuates the error that basic research is a luxury firms cannot afford. Reducing the scope of innovation in this way is a recipe for reducing innovation, period." (Lipartito, 2009, p. 153-154)

Introduction

The role of corporate research and central research laboratories1 has always been an important, and frequently controversial, subject in management science. Indeed Why do firms do basic research (with their own money) ? (Rosenberg, 1990) has been and remains a fundamental strategic question for academics, managers and policy makers. And actually we've seen important evolutions in the role of corporate research, from the apex of the 70's to the research blood bath [START_REF] Buderi | Engines of Tomorrow[END_REF] of the 90's and the current skepticism on the relevance of corporate research. This declining role of corporate laboratories over the last 40 years is now well documented [START_REF] Hounshell | The Evolution of Industrial Research in the United States[END_REF][START_REF] Mowery | Plus ça change. Industrial R&D in the 'third industrial revolution[END_REF][START_REF] Arora | The decline of science in corporate R&D[END_REF]. It has been notably (but not only) attributed to an isolation of research from corporate and business unit's strategy (e.g. [START_REF] Graham | The business of research. RCA and the VideoDisc[END_REF][START_REF] Graham | When the Corporation Almost Displaced the Entrepreneur: Rethinking the Political Economy of Research and Development[END_REF]. This paved the way for more « open » approaches to innovation processes [START_REF] Chesbrough | Open Innovation[END_REF][START_REF] Chesbrough | New Frontiers in Open Innovation[END_REF]. One of the consequences of this evolution has been a shift of academic research toward the analysis of interfirm organizations and networks to the detriment of the internal organization of research [START_REF] Powell | Interorganizational Collaboration and the Locus of Innovation: Networks of Learning in Biotechnology[END_REF][START_REF] Argyres | R&D, Organization Structure, and the Development of Corporate Technological Knowledge[END_REF]. This leaves us with the picture of corporate research as « dusty labs » (The Economists, 2007) that were « ivory towers » in which scientists defined their own objectives, most of the times loosely related to corporate strategy [START_REF] Buderi | Engines of Tomorrow[END_REF]. This perception of corporate research has been accentuated by the popular managerial literature. For example, the widely cited book Third Generation R&D by Roussel & al. (1991), that paved the way for 4, 5 or 6th generation (e.g. [START_REF] Nobelius | Towards the sixth generation of R&D management[END_REF], depicted the 50's and 60's, considered by some as the golden age of corporate research, as characterized by a « strategy of hope » in terms of management, i.e. investment on basic research that may (or not, with some luck) leads to breakthrough products. This view has gained momentum at the beginning of the 2000. And actually, behind the discourse, [START_REF] Arora | The decline of science in corporate R&D[END_REF] have documented a decline in the scientific content of industrial research, largely explained by the reluctance of large corporation to invest in research.

In this paper we want to question this story. Were the famous corporate research laboratories really the « ivory towers » depicted in the literature ? Incidentally, what are the conditions that makes corporate research strategically relevant ? Aren't we victim, as suggested by [START_REF] Lipartito | Rethinking the Invention Factory. Bell Laboratories in Perspective[END_REF], of an historical misperception of the role of corporate labs ?

To do this we adopt a qualitative methodology. We propose to revisit this question through an historical case study of Bell Laboratories. In particular we will focus on the post-war period and the invention of the transistor, which represent an archetypal success of corporate research leading to a breakthrough technology and the 1956 Nobel Prize in physics.

In so doing we will demonstrate that, contrary to a widely shared belief in the managerial literature, Bell Labs were completely integrated in ATT's strategy and part of carefully designed innovation process organizing a rapid transition from research to development and manufacturing, while simultaneously allowing the pursuit of the research effort. Thereby two different transistor design were invented at Bell Labs in 1947-48.

The paper is organized as follow. In the first part we present the evolution of corporate research and poses the research question. Section 3 present the methodology. The Bell Lab Case and the transistor story are presented in section 4. Section 5 discuss the case in the light of corporate research management. A conclusion summarizes the results.

The evolution of corporate research : an overview.

The history of corporate research is well documented in business history and innovation management (see [START_REF] Hounshell | The Evolution of Industrial Research in the United States[END_REF][START_REF] Mowery | Plus ça change. Industrial R&D in the 'third industrial revolution[END_REF][START_REF] Lecuyer | Manager l'innovation[END_REF][START_REF] Graham | When the Corporation Almost Displaced the Entrepreneur: Rethinking the Political Economy of Research and Development[END_REF], at least in the US case. In order to set the context of our reflection we can identify four main phases :

1. Corporate research as we know it appears in the US in the early 20th century, inspired by the german model, when firms like GE, DuPont, Westinghouse, Kodak or ATT created labs to internalize research in order, first, to control production processes and quality and, rapidly, to foster innovation. This leads to a tremendous growth of the number of research labs, from 50 in 1913 to 2000 in 1940 [START_REF] Lecuyer | Manager l'innovation[END_REF]. This widespread investment in research was justified by still emblematic successes like tunsgten filament lamps at GE (1909), vacuum tubes at Bell Labs (1912) and, probably the most famous, Nylon at DuPont (1935, see the classic from [START_REF] Hounshell | Science and Corporate Strategy[END_REF].

2. But this was only the beginning. The strategic role of research was indeed greatly amplified by the 2 nd world war. The fundamental role played by research during the conflict (see [START_REF] Buderi | The Invention that Changed the World[END_REF] for the radar project and Rhodes, 1986 for the atomic bomb/manhattan project), its legitimization by V. Bush famous Science : the endless frontier (1945) report2 , and the beginning of the Cold War opens the era of Cold War Science. The number of researchs labs exploded to 11000 in 1975 (Lécuyer 2015), following the massive impetus of defence-related spendings which, for example, represented 80% of federal R&D spendings in the 50's. This era was also marked by tremendous successes, typically the transistor (see below) or the great space-military projects (Atlas/Titan, Polaris, SAGE, Apollo, etc.).

3. However things begin to change in the late 60's -early 70's. First, the weight of military spendings and the prestige of physics pushes toward more fundamental research and created a gap with business goals at a time where US competitors, first and foremost Japan, focuses on product development performance. Second the economic crisis of the 70's lead firms to a relative decrease of federal R&D spendings and a fundamental questionning of the firm's organization. The crisis of the large diversified corporation and the move toward a recentering on « core business » leads to outsourcing of many activities and had profound consequences for research. Indeed some studies began to question the relevance of basic research (project hindsight, 1969) and firms starts to diversify their sources of new ideas. In particular they began to rely more on academic research, whose role has grown with cold war science. In this perspective the 23 million dollars grant from Monsanto to Harvard in 1974 is considered a turning point. The logical consequence of this move was the cut in firms R&D budget which culminated in the research bloodbath of the early 90's when many laboratories were abruptly closed (see [START_REF] Buderi | Engines of Tomorrow[END_REF].

4. This evolution was, in our view, theorized by H. Chesbrough in his famous Open Innovation (2003). He describes and analyzes the tranformation of research from a closed innovation model based on large central laboratories (mainly Xerox Parc in the book) to an « open » one where the firms largely relies on outside source of knowledge during the innovation process. This correspond to the tremendous growth of academic-industry partnership, interfirm cooperation, venture capital or the rise of consortium like sematech for semiconductors (1987). Biotech and ICT constitutes the paradigmatic cases of this evolution (for biotech see [START_REF] Powell | Interorganizational Collaboration and the Locus of Innovation: Networks of Learning in Biotechnology[END_REF][START_REF] Pisano | The evolution of science-based business: innovating the way we innovate[END_REF].

Wether this open innovation model is really new is a question that we leave open to debates (see [START_REF] Mowery | Plus ça change. Industrial R&D in the 'third industrial revolution[END_REF][START_REF] Chesbrough | Open Innovation[END_REF]Bogers 2014 reply).

This (roughly summarized) evolution has profound implication for the research role and management and for its place in academic research.Three points are worth noting 1. First the decline of corporate research [START_REF] Buderi | Engines of Tomorrow[END_REF] provides an excellent overview) leads to a corresponding decrease of firm's investment in science, R&D being more concerned by the Development and practical application than by the Research, i.e. the creation of new knowledge [START_REF] Arora | The decline of science in corporate R&D[END_REF].

2. This leads to a corresponding decline in the academic interest for Research as a specific function distinct from Development. During this period we observe a shift of academic research toward the analysis of interfirm organizations and networks to the detriment of the internal organization of research [START_REF] Powell | Interorganizational Collaboration and the Locus of Innovation: Networks of Learning in Biotechnology[END_REF][START_REF] Argyres | R&D, Organization Structure, and the Development of Corporate Technological Knowledge[END_REF]. Indeed, whereas dozens of papers deals with networks, firms cooperation or industry-academia partnership, the role of corporate research is fading out of view. This trend is quite clear in the following graph from Google Ngram which mirrored the evolution described above.

In this perspective it is striking to note that contemporary handbooks of innovation management does not deal with research. For example there is nothing on this topic in the Oxford Handbook of Innovation management (Dodgson & al, 2014) This historical overview leave us with the picture of corporate labs that were or became « ivory towers » in which scientists defined their own objectives, most of the times loosely related to corporate strategy (the so-called « strategy of hope » of the 50'S -60's in Roussel & al., 1991).

In this perspective investing in R&D is « fatalistic » i.e. mainly a cost that sometimes pay off with a bit of luck… This is reinforced by the focus of the literature on famous failures of this labs, for example RCA's Videodisc crash [START_REF] Graham | The business of research. RCA and the VideoDisc[END_REF] or the inability of Xerox to commercialize the invention of its much celebrated Xerox Park, most notably the personal computer [START_REF] Smith | Fumbling the Future. How Xerox Invented, Then Ignored, The First Personal Computer[END_REF][START_REF] Hiltzik | Dealers of Lightning. Xerox Parc and the Dawn of the Computer Age[END_REF]opening chapter of Chesbrough, 2003) 3 .

In this paper we want to question this perception of the role corporate research. Indeed we can wonder if the famous corporate research laboratories really were the « ivory towers » depicted in the contemporary literature ? Incidentally, this should lead us to discuss the conditions that makes corporate research strategically relevant. This is all the more important that, as pointed out by [START_REF] Graham | When the Corporation Almost Displaced the Entrepreneur: Rethinking the Political Economy of Research and Development[END_REF], the organization of research remains an important managerial question and its decline over the last decades may only be a a period in history. Important firms (Apple, Google…) are still investing in corporate research. In this perspective this work may yield important insight for the future management of Research.

Research methodology

Our approach is qualitative. We propose to revisit this question through an historical case study of Bell Laboratories during M.J. Kelly's tenure. More specifically we will focus on the invention of the transistor (roughly from the mid-30's to the mid-50's), which represent an archetypal success of corporate research leading to a breakthrough technology and the 1956 Nobel Prize in physics. Indeed we think that there is a lack of micro-analysis of the internal organization of research in the contemporary academic literature. Most of the (too rare) remaining literature on this question frequently relies on quantitative methods, for example to assess the relevance of different structural configurations (e.g. [START_REF] Arora | Make, Buy, Organize: The Interplay Between Research, External Knowledge, and Firm Structure[END_REF]. By focusing on a case of the post-war period, described as the « strategy of hope » phase, we want to question what we consider the myth of isolated research.

To do this, we rely on the historiography of Bell Labs and the transistor which, as far as we know, as not been fully exploited in economics and management research since R. Nelson's 1962 paper. Three sources are important for our purpose : 1) history of Bell Labs in general [START_REF] Lipartito | Rethinking the Invention Factory. Bell Laboratories in Perspective[END_REF][START_REF] Gertner | The Idea Factory: Bell Labs and the Great Age of American Innovation[END_REF] ; 2) history of the transistor [START_REF] Nelson | The Link Between Science and Invention: The Case of the Transistor[END_REF] ; but first and foremost [START_REF] Riordan | Crystal Fire. The Invention of the Transistor and the Birth of the Information Age[END_REF] ; 3) testimony of the actors involved in the process, in particular W. [START_REF] Shockley | Transistor technology evokes new physics[END_REF][START_REF] Shockley | The Invention of the Transistor -'An Example of Creative-Failure Methodology[END_REF][START_REF] Shockley | The Path to Conception of the Junction Transistor[END_REF], J.A. Morton (1970) and Bell Lab President, M.J. [START_REF] Kelly | A first Record of Thoughts Concerning an Important Postwar problem of the Bell Telephone Laboratories and Western Electric[END_REF][START_REF] Kelly | The Bell Telephone Laboratories-an example of an institute of creative technology[END_REF]. This testimonies are of particular significance since they question the common discourse of research as an « ivory tower » and the classic basic/applied distinction. This is particularly true of M. Kelly, Bell Labs President until 1959, who spent most his 1950 adress to discuss the systemic dimension of innovation and the corresponding Bell Labs organization.

This leads us to construct a case study of the transistor which is presented in the next section.

Bell Labs and the invention of the transistor

What were Bell Labs ? A short overview Our goal here is not to propose a complete history of Bell Laboratories which would be outside the scope of this paper (for an excellent overview see [START_REF] Gertner | The Idea Factory: Bell Labs and the Great Age of American Innovation[END_REF]. More simply we want to clarify what Bell Labs really were. Indeed, because of their outstanding successes (in particular 8 Nobel Prizes), they are frequently assimilated to a basic research organization. But this is wrong. Created in 1911 and formally as a subsidiary of ATT in 1925, Bell Labs first major success was, in 1912, the improvement of the vacuum tubes originally invented by Lee de Forest in 1906 that play a fundamental role to amplifiy the signal in long-distance communication, a strategic activity for ATT. This first success leads ATT to transform Bell Labs into a subsidiary in 1925. But their role has never been limited to research. They were a true research and development organization. They were thus in charge of research and engineering. And, when you look at the figures, research was actually a minor part of the activity of the 10 000 researchers and engineers, approximately 15% (Morton, 1972), the rest being devoted to system engineering (15%) and development (70% : 20% fundamental development / 50% specific development design). Therefore they took in charge the development of the new technologies and of the equipment necessary to « improve communication » in the huge ATT network, from customers terminal to submarine cable to long-distance amplifier, etc. In order to do so, they created carefully designed processes to manage the transition from research to fundamental development to specific development to implementation at Western Electric, the manufacturing arms of ATT, and in the Bell Divisions.

It is thefore misleading to assimilate the Bell Laboratories to a research organization. They were much more than that.

Inventing the transistor at Bell Labs 4

The story of the invention of the transistor at Bell Labs starts in this context at the beginning of the 30's. At this date Mervin J. Kelly 5 , Director of research, was well aware of the fundamental limitations of vacuum tubes, which played a fundamental role in telephone networks. It was clear at this date that vacuum tubes were fragile, hard to manufacture, power hungry and that they produce too much heat. Given their role as amplifier or electronic switches, this constitute a strategic concern for BL and ATT. One of the avenue identified by M. Kelly to overcome these weaknesses was the exploration of the potential of semiconductors (SC) 6 . The problem was that semiconductors physics was a nascent field at this time. Even if they were currently in use in some electronic equipment like radio and radar, semiconductors were hard to produce, difficult to experiment with and the theoretical knowledge of their inner functioning was very limited. Quantum physics, itself a new theoretical field in physics, has provided insights into their behavior but on very limited type of semicondictor (copper oxyde) and this remain a domain of fundamental research.

Given the novelty of the field, exploration at Bell Labs starts in a very academic manner. In 1936 M. Kelly decided to hire new brilliant PhD in physics to study the potential of SC for ATT, in particular the future Nobel prize winner William Shockley. A seminar was organized at BL to diffuse the knowledge of quantum physics among the scientists. W. Brattain, co-winner of the 1956 nobel prize and a famous experimenter was among them. This leads to the first experiments which tried to copy the structure of vacuum tubes in SC. The first try in copper oxyde in 1939 was a failure.

Word War II abruptly stops this research at BL. However the organization became deeply involved in a project that will have a major impact on the field of SC : the radar project (see [START_REF] Buderi | The Invention that Changed the World[END_REF]. Indeed vacuum tubes are too big and fragile for radars. Therefore they will be replaced by SC, in particular Germanium and Silicon. The war effort thus leads to tremendous improvements in the manufacturing of these SC. It also demonstrates the extraordinary impact of interdisciplinary research organized by project (Galison, 1997). The hypothesis was considered rich enough to lead Shockley to dedicate Bardeen & Brattain to a systematic exploration of the way to bypass surface states. The theoretician / experimenter duo then fully exploit the potentiel of the SC group to study a vast array of potential possibilities : doping, photovoltaic effect, magnetism, temperature, etc. 18 months later, in november 1947, Bardeen & Brattain confirms the surface states effect and show how to overcome them by using electrolyte.

This, according to [START_REF] Shockley | The Invention of the Transistor -'An Example of Creative-Failure Methodology[END_REF], marks the beginning of the magic month which will lead, in a burst of inventive scientific activity, to the invention of the transistor. Actually Bardeen & Brattain continue their work. Brattain suggestion to try the old « point contact » technique leads them to demonstrate a first amplification when the device is immersed in electrolyte (21/11/1947). It was too weak to be of any use, but it confirms the potential of this path. Then, on december 8, during a lunch conversation between Shockley, Brattain and Bardeen, the later suggest to switch from silicon, their material of choice, to the so-called « high back voltage germanium » produced at Purdue University during the war. They continue their test with different configurations (see [START_REF] Shockley | The Invention of the Transistor -'An Example of Creative-Failure Methodology[END_REF][START_REF] Hoddeson | The Discovery of the Point-Contact Transistor[END_REF] for a detailed presentation) until the famous december 16, 1947 experiment, during which they obtained a significant amplification. The first « transistor », called a « point-contact transistor », was born. It was announced publicly in june 1948.

However the story was far from over. The december 1947 device was a research prototype, very far from a working transistor. Furthermore, even if it works, the causes of the amplification were not understood by the team. Thus, in late 1947 -early 1948 the work goes 7 Nobel Prize 1963.

in two direction. First a « fundamental development » group, under the leadership of Jack Morton, was created to design a working device based on the december experiment. Second, W. Shockley, frustrated by not being one of the inventor 8 of the new device, decided to continue the exploration of amplification in SC by studying another design for the transistor. Indeed he was not completely convinced by the hypothesis that amplification happens at the surface of the SC in the point-contact transistor. Furthemore he remains a strong proponent of a « junction » device composed of different « layers » of SC. His frantic, lonely, theoretical works quickly pay off since, in late january 1948, he proposed a new design, the junction transistor with a n-p-n structure, based on a new hypothesis concerning amplification in SC, « minority carrier injection » 9 . Then comes a fundamental experiment at Bell Labs. While exploring the functioning of the point-contact transistor, JN Shieves, member of the SC group, decided to test a design in which the two contacts were placed on both sides of the SC, instead of close to one another. To his great surprises, the device works. This has a direct consequence : amplification does not happen at the surface of the device, as suggested in particular by Bardeen. However, when he presents his results to the group, he explains that he had no alternative explanation. This is the moment chosen by Shockley to unveil to the stunned group his new theory that explains both the phenomena observed by Shieves and the potential of the junction transistor. It will took another two years, the transfer to « exploratory development », and fundamental breakthrough in the metallurgy of SC, to demonstrate the feasibility of a junction transistor.

The epilogue of the story is well-known (see Brown & MacDonald, 1982 ;[START_REF] Lecuyer | Making Silicon Valley. Innovation and the Growth of High Tech, 1930-1970[END_REF]. The point-contact transistor proved to be an industrial nightmare and had only limited application, mainly in the military. The junction transistor, on the contrary, becomes the foundation of the electronic revolution. W. Shockley directly triggered this revolution when he leaves BL to create his own firm, Shockley Semiconductor Laboratory (SSL), in Palo Alto in 1956, the year he receives the Nobel Prize in physics with Bardeen & Brattain. Even if his firms failed, he is « the man who brought silicon to the valley » and SSL was an extraordinary school for the future leaders of the semiconductor industry (G. Moore & R. Noce, typically) that created Fairchild Semiconductor in 1957, after leaving SSL. Here again military applications, first and foremost the Minuteman ICBM program, were key to the take off of this industry. On the ATT side, it will take more years to semiconductors to start replace vacuum tubes and mechanical 8 « Frankly Bardeens and Brattain's point-contact transistor provoked conflicting emotions in me. My elation with the group's success was balanced by not being one of the inventors. » in W. Shockley, 1974, p. 54. 9 « Minority carrier injection, in electronics, a process taking place at the boundary between p-type and ntype semiconductor materials, used in some types of transistors. Each semiconductor material contains two types of freely moving charges: electrons (negative charges) and holes (positive charges). Electrons are the more abundant, or majority, carrier in n-type materials, holes being the less abundant, or minority, carrier. In p-type materials, however, holes are the majority carrier, and electrons the minority carrier. If a battery is properly connected to the semiconductor material, the p-type material may acquire additional electrons (minority carriers), injected into the p-type material from the n-type material by the flow of electrons from the battery. This is minority carrier injection. It is important in bipolar junction transistors, which are made of two p-n junctions. » Source : https://www.britannica.com/technology/minority-carrier-injection in the Bell System10 . Computer will of course be the driving application of transistors but, legaly forbidden to enter this market for anti-trust reasons, ATT did not benefit from their extraordinary development.

Analysis : Science and corporate strategy at Bell Labs

The transistor case demonstrate that, contrary to a widely shared belief in the managerial literature, Bell Labs were very carefully integrated in ATT's strategy. It was not the result of a « strategy of hope » i.e. of a research decided by researcher themselves that, almost accidentally, leads to a breakthrough innovation. Quite the contrary. Indeed, as we have seen, the invention of the transistor is the result of a research effort launched in the mid 30's by M. Kelly on the basis of a thourough assessment of limits of current technology (vacuum tubes) and of ATT future challenges (high volume / high speed trafic). Moreover this commitment of the firm lasted over 10 years (15 if we consider the war period). In this perspective, as demonstrated by Lipartito ( 2009) M. Kelly follow the strategy of his predecessors, in particular F. Jewett, of a research organization carefully integrated in ATT strategy and processes. Indeed, from their creation, BL had developed processes to develop strong link with strategy on one side, and with manufacturing on the other. It has never been « isolated » from ATT. As explained by Lipartito, « here was the lab strengh, « continuous operation » from research to application. Though personnel were free to pursue fundamental work, the labs was not set up « separate and apart » from daily operations of « commercial design and economic consideration », as were other research entities. Research and development department were in « close proximity », and information flowed between them casually and informally. The research worker served as a consultant to the development engineer, and researchers had a good understanding of the field operations of the apparatus they are working on ». (2009, p. 144).

In this perspective it is very interesting to note that, in a 1943 memo entitled A first Record of Thoughts Concerning an Important Postwar problem of the Bell Telephone Laboratories and Western Electric 11 , M. Kelly, now Director of research, discuss at lenght what he considers the main problem of BL i.e. … the link with manufacturing !! His concern was to smooth the transfer from research to production and to help the division move to the new electronic technologies developed at the lab. This leads to the creation of branch laboratories within the divisions. This trend was reinforced later by the creation of a department of system engineering at Bell Labs in 1955 in order to integrate their work in the ATT network 12 . This emphasis on the integration of research in ATT's innovation process, is clearly visible in the transistor case with the swift move from research to exploratory development. Research at Bell Labs was was part of a carefully designed innovation process organizing a rapid transition from research to development and manufacturing, while simultaneously allowing the pursuit of the research effort.

The invention of the transistor was also, a point rarely mentioned in the literature, strongly influenced by the new practices of research/engineering management developed in major word war II projects, such as the Radar (in which Bell Labs plays a major role) and Manhattan projects. It benefited from M. Kelly's reorganization of Bell Labs after the war, namely the creation of an interdisciplinary research group on semi-conductors, under the leadership of W. Shockley. Indeed interdisciplinarity foster a very rich, multidimensional, exploration process and facilitates rapid experimentation capability (see [START_REF] Thomke | Experimentation Matters[END_REF]. The magic month, as Shockley called it, constitutes a spectular example of the relevance of this approach (see below). Here again research management plays a central role.

The magic month measured by scientist's notebook entries in [START_REF] Shockley | The Invention of the Transistor -'An Example of Creative-Failure Methodology[END_REF] What is very interesting for our research question is to discuss the impact of this policy on the researchers themselves. Here the recollections of W. Shockley are very interesting. He is crystal clear that M. Kelly's policy had a major impact on his work and motivation right from the start. For example, in his 1976 recollection of the invention of the transistor [START_REF] Shockley | The Path to Conception of the Junction Transistor[END_REF] he always make the link between his scientific accomplishments and Kelly's initial vision explainging that « A key motivation that stimulated my will to think about transistors came from Dr. M. Kelly, who

Discussion and Conlusion

This journey through Bell Labs and the invention of the transistor leads us to question the prevalent discourse of corporate research. In particular it refute the conventional wisdom, so frequent in the managerial press, that corporate research was an kind of ivory tower in the postwar period. Quite the contrary actually. We can even wonder if the Roussel & al's, « strategy of hope » really existed in the 50-60's ? In so doing we concur with K. Lipartito when he affirm that :

"The current skepticism about basic research in industry imagines a world of detached corportate laboratory that existed only for a moment in a much longer history of 13 Our emphasis.

successful balancing of the commitment to the long term with the need for a commercial payoff. This historical misperception perpetuates the error that basic research is a luxury firms cannot afford. Reducing the scope of innovation in this way is a recipe for reducing innovation, period." (Lipartito, 2009, p. 153-154) What we have here, on the contrary, is a model of strategic research management in which the central laboratory is carefully integrated in the firms innovation process, from strategy formulation to product launch. In this perspective, the reading of J.A. Morton's Organizing for innovation published in 1971, is fascinating. J. Morton was responsible for the fundamental development group in charge of developing the transistors for commercial applications, before becoming VP for electronics technology at Bell Labs. He spends the entire book, based on his Bell Labs experience, demonstrating the relevance of a system approach to innovation and analyzing how relationship between BL and ATT were carefully organized from research to sales and services. This is a significant reminder since, as pointed out by [START_REF] Graham | When the Corporation Almost Displaced the Entrepreneur: Rethinking the Political Economy of Research and Development[END_REF], the organization of research remains an important question. Its decline over the last decades may only be a a period in history. Important firms (Apple, Google…) are still investing in corporate research. In this perspective this work brings us back to the future by debunking the myth of isolated research. It underlines that internal research may remains a powerful way to overcome the challenges of risk, integration and learning typical of science-based businesses [START_REF] Pisano | The evolution of science-based business: innovating the way we innovate[END_REF].

In this perspective, it outlines a model of research management14 :

anchored in the firm's strategic vision, which help to define innovative design spaces to be explored, even broadly such as « solid state physics to understand SC behaviors for telecom applications » inserted in the overall innovation process, based on multidisciplinary, mission-oriented teamwork and, last but not least, not enslaved to development but capable of being, at the same time, focused on pratical application and on generic theoretical knowledge [START_REF] Shockley | Electrons and Holes in Semiconductors with applications to transistor electronics[END_REF]. This last point deserves an explanation. It is interesting to note that BL make three different moves in december 1947 1. setting-up of a « fundamental development » group to start the transition to a commercial product ; 2. continuation of the research on the point-contact transistor under the leadership of Bardeen & Brattain ; 3. W. Shockley's (at first secret) work on the junction transistor. This means, and this constitutes a very interesting way to manage research, that they were able to simultaneously develop commercial product, explore other alternatives for SC and increase of the K base on the generic properties of SC. Indeed, as a result of his work, Shockley publishes Electron and holes in semiconductors with application to transistor electronics in 1950, which constitute the first codified knowledge base for the nascent semiconductor industry.

Therefore what we have here echoes the recent work on the management research, in particular Le Masson & Weil (2016) on « conceptive » vs. problem-oriented research. In the first case the first role of research is to speed-up product development, to solve the problem raised by engineering. On the contrary « conceptive » research emphasizes that a major goal of research is to explore unknown spaces (e.g. solid-state physics for SC) and to do this by designing generic laws and properties of, in our case, a class of material in order to increase their relevance for a wide range of applications. Note, as explained by Shockley and demonstrated by the transistor case, that this is not contradictory with the simultaneous development of pratical applications and the integration of research in the innovation process.

or in the Handbook of New Product Development (Loch & Kavadias, 2008). Even the standard Strategic Management of Technology and Innovation (Burgelman & al, 5th edition

  , 2008) comprise only 3 papers on this question (and quite old one) on the 88 papers composing the book. A complete literature review remains to be done but a first search in academic journals (CIM, Research Policy, ICC, JPIM, SMJ in particular) leads us to hypothesize that this shift toward open innovation is also present in peer-reviewed journals.

3. The trend is even clearer in the economic press which eagerly celebrates the end of central R&D in the early 2000's. An complete overview is impossible but the trend is clear in articles such as « Is the Central R&D Lab Obsolete? » (MIT Technology Review, april 24, 2001), « Why Big Companies Can't Invent ? », (MIT Technology Review, may 1, 2004) « Corporate Labs Disappear. Academia Steps In ». (New-York Times dec 16, 2007) or « The rise and fall of corporate R&D. Out of the dusty labs. » (The Economist, Mars 2007).

4

  This section is based on an ongoing work with L. Petitgirard (CNAM). SeeLenfle & Petitgirard, 2018. The fundamental reference on this story areHoddeson, 1981 and Riordan & Hoddeson, 1997. 5 Mervin J.Kelly was Director of Research for 1936-1944, Executive Vice-President for 1944-1951, and President of Bell Labs for 1951-1959. 6 « A semiconductor material has an electrical conductivity value falling between that of a metal, like copper, gold, etc. and an insulator, such as glass. Its resistance decreases as its temperature increases, which is behaviour opposite to that of a metal. Its conducting properties may be altered in useful ways by the deliberate, controlled introduction of impurities ("doping") into the crystal structure. (…) Some examples of semiconductors are silicon, germanium, and gallium arsenide. Semiconductor Rich from this experience, research starts again at BL in april 1945. W. Shockley & W. Brattain, back from their war duties, experiments again with a new design based on the so-called « field effect ». This was a new failure. But soon the organization changed. From his own experience of the war, specifically the radar project, M. Kelly decided to create a multidisciplinary team to study solid-state physics. Under the leadership of Shockley, the team comprises experimenters, theoreticians, chemists, electronician, etc. They were soon co-located on BL new campus build at Murray Hill, NJ. They were driven by the goal defined by Kelly in the mid-30's : design of amplifier in SC for applications in ATT's telephone network. Their first decision, based on the knowkedge gained during the conflict, was to focus the research on germanium and silicon. Indeed, the radar project had demonstrated their potential, even if they were not well understood theoretically. Simultaneously the SC team continues to grow. October 1945 was thus marked by the arrival of J. Bardeen, who had just finished his PhD at Princeton on Quantum physics under the supervision of E. Wigner 7 . Shockley asked him to check the validity of his calculation to understand the failure of the april experiments. Bardeen confirm the calculation and his research leads him to propose, in march 1946, that the failure may be caused by a « surface states » effect that « trapped » the electron at the surface of the material.

(…)

Semiconductor devices can display a range of useful properties such as passing current more easily in one direction than the other, showing variable resistance, and sensitivity to light or heat. Because the electrical properties of a semiconductor material can be modified by doping, or by the application of electrical fields or light, devices made from semiconductors can be used for amplification, switching, and energy conversion. » Source : https://en.wikipedia.org/wiki/

  was then Director of Research at BL, a position he held before becoming President some years later. Dr Kelly visited me for the purpose of emphasizing his objective of introducing electronic switching in the telephone system. He said that he looked forward to the time when metal contacts, which were used in telephone exchanges to make connections when numbers that are dialed, would be replaced by electronic devices. His interest in his goals was very great. He stresses its importance to me so vividly that it made an indelible impression. » Later in the same paper when explaining the patent for the junction transistor he explains that « The title « high power large area semiconductor valve » of the disclosure was not prophetic. The great contribution of the junction transistor has not been its high power handling capacity. Quite the contrary. The good high frequency performance at unprecedented low power level is what revolutionized the computer industry. This revolution has included electronic switching in telephone exhanges, the objective that Mervin Kelly had emphasized to me with such enthusiasm during my first years or so at BL 13 . ».This, this should be noted, draw the outline of an original model of research management in which the distinction between fundamental et applied research vanishes. His 1956 Nobel lecture is very interesting on this question since it starts with an appeal to go beyond the basic / applied dichotomy (on this question seeNaryanamurti & Odumosu, 2016).

« The objective of producing useful devices has strongly influenced the choice of the research projects with which I have been associated. It is frequently said that having a more-or-less specific practical goal in mind will degrade the quality of research. I do not believe that this is necessarily the case and to make my point in this lecture I have chosen my examples of the new physics of semiconductors from research projects which were very definitely motivated by practical onsiderations. » (W.

Shockley, Nobel Lecture, 1956)

.

In this paper with discuss specifically the role of research, as distinct from developpement. In order to simpfily let's say that Research focuses on the creation of new knowledge, whereas development takes this knowledge to the product launch phase through engineering.

Criticizing Bush report is outside the scope of this paper. Let's just say that we agree with Naraynamurti & Odumosu (2016) when they point to a misunderstanding by Bush of the causes of the radar project success which as much basic research as it was engineering.

This focus on the Xerox PARC is actually questionable. Indeed the PARC was 1) created quite late in this story(1970) ; 2) from scratch by hiring people outside of Xerox ; 3) didn't play any role in Xerox history (contrary to its Webster research center) before 1970. Therefore the PARC is not representative of the historical corporate research centers like Bell Labs, for example (created in 1912 and in 1925 as a subisidiary).

The first application was in a card translator equipment in 1952[START_REF] Kelly | The First Five Years of the Transistor[END_REF]. They appear in transatlantic cable as a substitute to vacuum tubes in… 1968(Morton, 1970).

We are very grateful to Dr. S. Hochheiser, from ATT Archives in Warren, NJ, for sending us this document.

See also[START_REF] Kelly | The Bell Telephone Laboratories-an example of an institute of creative technology[END_REF] adree to the Royal Society in London in which in explain at lenght the integration of research in the overall innovation process. It is interesting to note the place granted to the physical organization of the lab to foster interdisciplinary cooperation and flexibility.

Bell Labs are not the only case of this management of research. See Le Masson & al. (2010) for analysis of the famous Nylon / DuPont case.


Enrico Mingo Hoffman

Sai Kishor Kothakota

Adria Roig Moreno

Andrea Curti

Narcis Miguel

Luca Marchionni

Whole-Body Kinematics Modeling in presence of Closed-Linkages: application to the Kangaroo Biped Robot

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

The presence of closed and parallel linkages in robots have multiple advantages w.r.t. open-kinematic chains, e.g. the possibility to relocate actuation to achieve better mass and inertia distribution, superior stiffness and high payload-to-weight ratio [START_REF] Kumar | A survey on modularity and distributivity in series-parallel hybrid robots[END_REF]. Despite these clear advantages, open-kinematic chains remain a popular choice, particularly in legged robotics, due to simpler design and control. However, closed linkages are becoming more and more used in the new generation of biped robots, in particular to achieve highly dynamic movements with high resilience to impact. In fact, the new BostonDynamic Atlas [START_REF] Guizzo | By leaps and bounds: An exclusive look at how boston dynamics is redefining robot agility[END_REF] and AgilityRobotics Cassie/Digit [START_REF] Hurst | Walk this way: To be useful around people, robots need to learn how to move like we do[END_REF], present closed linkages in their mechanical design. In particular, Cassie/Digit uses electric actuators localized near its torso/pelvis with a transmission, consisting of closed-bar linkages, to transfer the motion in order to reduce inertia, especially during impacts while walking.

Following this new trend in humanoid robotics, in this work we present preliminary studies on the kinematic modeling and control of the legs of the Kangaroo platform, a novel bipedal platform recently designed and developed at PAL Robotics. Kangaroo follows a series of PAL humanoid robots, started back in 2004 with the REEM-A, passing through the REEM-B and C series, and TALOS [START_REF] Stasse | Talos: A new humanoid research platform targeted for industrial applications[END_REF] platform, as shown in Figure 1. The Kangaroo robot has been designed using novel high-power and robust linear actuation units located near the pelvis area, while the motion is transferred to joints through a complex system of closed parallel linkages. This design entails multiple advantages such as robustness during impact, low inertia at the end-effectors and light-weight leg structure, making Kangaroo the ideal platform to perform highly dynamic and contact All the Authors are with PAL Robotics, Carrer de Pujades 77, 08005 Barcelona, Spain. E-mail: {enrico.mingo, sai.kishor, adria.roig, andrea.curti, narcis.miguel, luca.marchionni}@pal-robotics.com rich movements. We introduce how we modeled the legs of Kangaroo and how to encompass this complex systems of closed linkages into Quadratic Programming (QP)-based Whole-Body controllers.

This short paper is organized as follows: Section II briefly overviews the related works, Section III introduces the Kangaroo platform and its modeling, Section IV presents an approach to QP-based whole-body control considering both actuated and passive DOFs as optimization variables, Section V presents results of the proposed approach and finally Section VI discusses conclusions and future works. II. RELATED WORKS Several bipedal humanoid robots present closed linkages, in particular parallel four-bars mechanisms for the actuation of the pitch-roll ankle joints [START_REF] Knabe | Team valor's escher: A novel electromechanical biped for the darpa robotics challenge[END_REF]- [START_REF] Radford | Valkyrie: Nasa's first bipedal humanoid robot[END_REF]. However, in this case the actuators are moved along the shin, making the advantages of such designs localized only to the ankle part. Also, concerning the control, most of the time the mechanism is considered isolated from the full-model of the robot and the joint side is treated as actuated while performing control, possibly introducing errors when considering linear joint limits.

Due to the recent development of bipedal legged system with closed linkages, several works on the control and modeling side are emerging. For example [START_REF] Carpentier | Proximal and sparse resolution of constrained dynamic equations[END_REF], proposes a novel efficient method for solving forward dynamics equations involving constraints, such as closed linkages. In [START_REF] Schumacher | A versatile inverse kinematics formulation for retargeting motions onto robots with kinematic loops[END_REF], a robust Inverse Kinematics (IK) algorithm for robots with closed linkages is proposed.

The purpose of this short paper is to describe how the complex closed linkages in Kangaroo legs are modeled and included inside a hierarchical whole-body QP-based controller.

III. THE KANGAROO ROBOT

The Kangaroo robot is actuated by 12 linear ball-screw modules with electrical motors, 6 per leg, while the complete leg can be modeled by 76 DOFs, among them, 64 are passive. The peculiar design of these linkages on Kangaroo permits to move the hip in the roll, pitch and yaw directions, to extend the feet maintaining constant orientation of the ankle w.r.t. the base frame emulating a prismatic joint, and to move the ankle in the roll and pitch directions. In particular, the extension/retraction of the leg is obtained by just actuating the motor on the femur. Notice that the hip pitch and roll, as well the ankle pitch and roll are coupled, by the respective parallel linkages.

To model such complex mechanical structure, we use a constraint-based formulation for mechanical joints where the closed linkage is opened at one joint and a constraint is added. Saying that, a single leg of Kangaroo can be divided into 4 parts, each one consists of one or more closed linkages opened at one specific joint (see Table I). Each spatial linkage, introduces a constraint in the form:

a p u (θ) = 0, (1)
with a p u (θ) ∈ R m the relative pose between the frame F a and the frame F u , as shown for the case of the crank slider in Figure 3, that differentiate became:

a J a,u (θ) θ = 0, (2)
with a J a,u (θ) ∈ R m×n the relative Jacobian between the frame F a and the frame F u , expressed in frame F a , and

θ = θ u θ a ∈ R n , (3)
where θ a ∈ R a are the actuated DOFs and θ u ∈ R u are the passive DOFs, such that n = a + u. In this work we consider mechanical closed linkages with no redundancy, therefore m = u.

Fig. 3: Crank Slider linkage.

Dividing the active from the passive part of the constraint Jacobian (1) is possible to derive a mapping Jacobian to map actuated velocities to underactuated ones:

θu = J m (θ) θa , (4)
and (virtual) underactuated torques to actuated ones:

τ a = J m (θ) T τ u , (5)
with J m ∈ R u×a . More complex linkages, such as differential and parallel linkages, can be modeled by just stacking the constraint Jacobian computed for each (sub-)closed linkage. For example, considering the spatial differential linkage in Figure 4, we consider the two Jacobians a1 J a1,u1 (θ) and a2 J a2,u2 (θ) computed considering half of the full differential linkage and stacking them such that:

J (θ) = a1 J a1,u1 (θ) a2 J a2,u2 (θ) . (6)
Notice that also in this case the number of introduced constraint matches with the number of passive DOFs. Therefore, the full set of linkages for the Kangaroo robot can be represented by the constraint equation ( 2) with n = 76, m = u = 64 and a = 12.

IV. QP-BASED KINEMATIC WHOLE-BODY CONTROL

A floating-base system in contact with the environment presents the same constraint depicted in (2), one per contact but expressed w.r.t. the inertial frame (see Figure 5): being q the robot generalized coordinates:

J c (q) ν = 0, (7)
q =   p ρ θ   , (8)
with p ∈ R 3 and ρ ∈ R 4 a parameterization in SE(3) of the pose of the floating-base, in particular using quaternions for the orientation, and θ ∈ R n the actuated and underactuated DOFs in the robot, therefore q ∈ R n+7 , ν ∈ R n+6 the generalized velocities:

ν =   ṗ ω θ   , (9)
and J c (q) = J T c,1 (q) J T c,2 (q) T ∈ R 12×n+6 , considering 2 planar contacts with the environment. State-of-the-art techniques for kinematic whole-body control of floating-base systems make use of QP optimization, permitting also to take into account priorities between Tasks, and the inclusion of Constraints, in the form of equalities and inequalities [START_REF] Hoffman | Robot control for dummies: Insights and examples using opensot[END_REF]. In the whole-body control framework, the closed linkage is another type of equality constraint in the form a J a,u (q) ν = λ a e u (q), (

) 10
with a e u (q) the error associated to the pose (1) and λ a positive gain. This term is used to satisfy exactly the closed linkage constraint once the solution of the QP, a generalized optimal velocity ν * , is integrated to obtain the new generalized coordinates q:

q k+1 = q k ⊕ dtν * (11)
at each control loop, with ⊕ a proper operator that integrates the floating-base angular velocities in SE( 3) and dt the control cycle time.

We will now present three optimization schemes to realize Forward Kinematics (FK), Inverse Kinematics (IK), and Cartesian Inverse Kinematics (CIK), considering actuation position limits and velocity limits1 .

A. Forward/Inverse Kinematics

We consider the problem of computing the passive DOFs positions θ u given a reference for the motor ones θ a,r , namely Forward Kinematics problem, and its counterpart, the problem of computing the active DOFs positions θ a given references for a sub-set of the passive ones θu,r ∈ R a , namely Inverse Kinematics problem.

Both problems can be solved using the same QP problem structure:

min ν ∥x -x r ∥ 2 + ϵ ∥ν∥ 2 (12a) s.t. a J a,u (q)ν = λ a e u (q) θ m -θ dt ≤ θ ≤ θ M -θ dt (12b) θa,m ≤ θa ≤ θa,M
with different meaning for the x and x r in the case of FK or IK, ϵ a regularization term, θ M and θ m the maximum and minimum DOFs position limits and θa,M and θa,m the actuated DOFs velocity limits. Notice that not all the passive DOFs position limits may be known while actuated DOFs position limits are normally known.

In particular, for the FK:

x = θa , x r = θa,r = K (θ a,r -θ a ) ,
while for the IK:

x = θu , x r = θu,r = K θu,r -θu ,
with K a positive definite diagonal gain matrix.

For the FK/IK problems, the presence of the floatingbase velocities is not really important, in fact the constraint Jacobian (2) does not depend on them.

B. Cartesian Inverse Kinematics

We consider the problem of Operational Space control in presence of contacts, in particular the position of the Center of Mass (CoM) and the orientation of the base of the robot.

We denote the CoM task as:

J CoM (q)ν = v CoM,r (q) ( 15
)
with v CoM,r (q) = K CoM (x CoM,r -x CoM (q)), K CoM a positive definite diagonal gain matrix, x CoM (q) the actual position of the CoM and x CoM,r the desired one.

Similarly, the orientation task for the floating-base:

J f b (q)ν = ω f b,r (q) (16)
with ω f b,r (q) = K f b e f b (q), K f b a positive definite diagonal gain matrix and e f b (q) a properly computed orientation error for the floating-base. Finally, for the feet/contacts task, we consider a modified version of (7):

J c (q)ν = K c e c (q), (17) encompassing as well the pose error e c (q) properly computed, with K c a positive definite diagonal gain matrix.

For the CIK we can consider priorities between feet/contact tasks and CoM and base orientation tasks. In particular, to have motions of the base that are contact consistent, the feet/contacts tasks has to be at high priority w.r.t. the CoM and base orientation tasks. The latter should therefore act on the null-space of the contacts.

We first setup a level 0 QP problem where we solve:

min ν ∥J c (q)ν -K c e c (q)∥ + ϵ ∥ν∥ s.t. constraints (12b) (18)
computing the optimal solution ν 0 .

We consecutively setup a level 1 QP in the form:

min ν ∥F CoM (q)∥ w1 + ∥F f b (q)∥ w2 + ϵ ∥ν∥ s.t. constraints (12b) (19) J c (q)ν = J c (q)ν 0
where the final optimal solution ν * is computed, with F CoM (q) = J CoM (q)ν -v CoM,r (q), F f b (q) = J f b (q)νω f b,r (q), and w (•) ∈ R 3×3 positive-defininte relative weights between the tasks. This second QP permits to track desired CoM position and floating-base orientation ensuring contact consistency.

It is worth to notice that putting directly the feet/contacts tasks as constraints, in other word using only the level 1 QP with J c (q)ν 0 = K c e c (q), may result in unfeasiblity due to incompatibility between the feet/contacts constraint and the other constraints. This technique of cascading QPs to enforce priorities is often called inequality Hierarchical QP (iHQP) [START_REF] Kanoun | Kinematic control of redundant manipulators: Generalizing the task-priority framework to inequality task[END_REF]. Both the QPs in ( 18) and ( 19) can be solved using state-of-the-art QP solvers such as qpOASES [START_REF] Ferreau | qpoases: A parametric active-set algorithm for quadratic programming[END_REF].

V. RESULTS

The proposed Whole-Body QP-based schemes, discussed in the previous Section IV, has been implemented using the CartesI/O framework [START_REF] Laurenzi | Cartesi/o: A ros based real-time capable cartesian control framework[END_REF]. Simulation results are reported in the video at the following link https://youtu.be/tl7o0jGMaOw.

Preliminary experiments on the real Kangaroo platform estimating and visualizing the whole pose of the kinematics structure of the robot using the mapping Jacobian in (4), for which we wrote a dedicated C++ library called Closed Linkage Library (CLL), see Figure 6, are reported in the second video https://youtu.be/Cc-WILUqlKE.

VI. CONLCUSION AND FUTURE WORKS

This short paper presented the approach to model the closed linkages in the legs of Kangaroo, the new humanoid platform developed by PAL Robotics, based on relative Jacobian. The computed closed linkage constraint is also used to derive a mapping Jacobian between actuated and underactuated quantities. The closed linkage constraint is added inside a QPbased whole-body kinematic controller in order to produce or estimate consistent motions of the legs. Future works will consider to extend the formulation to floating-base dynamics with a particular with a focus on exploiting actuated and passive structure of the closed linkage constraint in order to reduce computation complexity.

Fig. 1 :

 1 Fig. 1: Humanoid bipedal robots designed and produced by PAL Robotics since 2004.

Figure 2 5 ankleFig. 2 :

 252 Fig. 2: On the left, side and front view of the Kangaroo robot, on the right, side and front view of Kangaroo model with frames visualized in RViz.

  (a) Full differential linkage. (b) Half differential linkage.

Fig. 4 :

 4 Fig. 4: Differential linkage modeling.

Fig. 5 :

 5 Fig. 5: Floating-Base system in contact with the environment, presenting multiple closed chain linkages.

Fig. 6 :

 6 Fig. 6: FK mapping to estimate underactuation state for visualization purpose on the Kangaroo real hardware.

TABLE I :

 I Kangaroo leg's closed linkages

	Part	Type	#DOFs Passive Active
	Hip Yaw	Crank slider	2	1
	Hip Pitch/Roll	Differential	8	2
	Knee	Planar	6	1
	Ankle Pitch/Roll	Differential	16	2

2008 2013 2017 2022

From now on we will discard the dependency on q for conciseness.


David Colas

Cédric Bellis

Régis Bruno Lombard

Régis Cottereau

Crossover between quantum

published or not. The documents may come

Anderson localization refers to wave localization due to the presence of a strongly inhomogeneous medium. Originally predicted by P. W. Anderson in 1958 for electronic wave functions [START_REF] Anderson | Absence of diffusion in certain random lattices[END_REF], this phenomenon was since evidenced in a great variety of oscillatory systems, such as electromagnetic [START_REF] Laurent | Localized modes in a finite-size open disordered microwave cavity[END_REF][START_REF] Schwartz | Transport and Anderson localization in disordered twodimensional photonic lattices[END_REF][START_REF] Riboli | Anderson localization of near-visible light in two dimensions[END_REF] or matter waves [START_REF] Piraud | Anderson localization of matter waves in tailored disordered potentials[END_REF][START_REF] Billy | Direct observation of Anderson localization of matter waves in a controlled disorder[END_REF][START_REF] White | Observation of twodimensional Anderson localisation of ultracold atoms[END_REF], in the context of meta-materials [START_REF] Asatryan | Anderson localization of classical waves in weakly scattering metamaterials[END_REF], photonic lattices [START_REF] Lahini | Anderson localization and nonlinearity in one-dimensional disordered photonic lattices[END_REF] or cavity QED [START_REF] Sapienza | Cavity quantum electrodynamics with Anderson-localized modes[END_REF]. Anderson localization has also been studied for classical vibrating systems, mostly for ultrasounds [START_REF] Condat | Observability of acoustical and optical localization[END_REF][START_REF] Weaver | Anderson localization of ultrasound[END_REF][START_REF] Lobkis | Anderson localization of ultrasound in plates: Further experimental results[END_REF][START_REF] Hu | Localization of ultrasound in a threedimensional elastic network[END_REF][START_REF] Dhillon | Localization of ultrasound in 2d phononic crystal with randomly oriented asymmetric scatterers[END_REF]. Notable differences exist in the mathematical structure of the operators describing classical and quantum waves, and the general properties of Anderson localization, such as the link between the presence of spectral gaps and the emergence of localized modes, are still discussed [START_REF] Kirkpatrick | Localization of acoustic waves[END_REF][START_REF] Figotin | Localization of classical waves i: Acoustic waves[END_REF][START_REF] Altmann | Quantitative Anderson localization of Schrödinger eigenstates under disorder potentials[END_REF]. Despite this abundant literature produced over six decades, many questions on the nature of Anderson localization remain open. One important concern is the following: is it possible to determine, from the knowledge of the random medium configuration, where waves are going to localize? And this, without solving the computationally expensive associated eigenvalue problem.

In 2012, M. Filoche and S. Mayboroda developed an original and innovative tool to apprehend wave localization, which they coined as the localization landscape [START_REF] Filoche | Universal mechanism for Anderson and weak localization[END_REF]. For a given quantum potential V (x), possibly random, one considers the eigenvalue problem for the Schrödinger equation in a domain Ω

(-∆ + V (x))ψ n (x) = E n ψ n (x) , ψ n ∂Ω = 0 . (1)
Instead of Eq.( 1), one can solve the elliptic equation

(-∆ + V (x))u ψ (x) = 1 , u ψ ∂Ω = 0 , (2)
with u ψ the so-called Localization Landscape (LL).

When analyzing the LL's shape, i.e. its peaks and valleys, one can predict the position, shape and energy of * cottereau@lma.cnrs-mrs.fr the localized eigenstates, to a certain extent. The LL can actually be understood as the inverse of an effective confining potential for the eigenmodes [START_REF] Arnold | Effective confining potential of quantum states in disordered media[END_REF], and it has been successfully applied in various contexts [START_REF] Filoche | Localization landscape theory of disorder in semiconductors. i. theory and modeling[END_REF][START_REF] Piccardo | Localization landscape theory of disorder in semiconductors. ii. urbach tails of disordered quantum well layers[END_REF][START_REF] Li | Localization landscape theory of disorder in semiconductors. iii. application to carrier transport and recombination in light emitting diodes[END_REF][START_REF] Chalopin | Universality of fold-encoded localized vibrations in enzymes[END_REF][START_REF] Balasubramanian | Manybody localization landscape[END_REF][START_REF] Lemut | Localization landscape for dirac fermions[END_REF][START_REF] Garcia | Localized modes prediction in a membrane with nonuniform tension from the quasi-static measurement of its localization landscape[END_REF][START_REF] Shamailov | Computing the eigenstate localisation length at very low energies from localisation landscape theory[END_REF]. This remarkable theory is also extendable to localization associated with other symmetric elliptic differential operators governing wave propagation, such as the Laplacian, the bilaplacian or the operator -div(A(x)∇), notably describing classical waves in inhomogeneous media. However, as we will show, the LL only returns useful information in the case of low frequency localization. When the system exhibits high frequency localization, i.e. when the first eigenmodes are delocalized, the LL does not bring any insight on the position or shape of the localized modes. Such situation might occur upon the choice of the differential operator or the boundary conditions.

In this Letter, we adapt the Localization Landscape Theory (LLT) to classical scalar waves exhibiting high frequency localization, using the example of a classical wave equation such as the Helmholtz equation governing acoustics. For a given type of random quantum potential or acoustic structure, we first review and compare the phenomenology associated with the quantum and classical systems, which is essentially an opposite transition between the localization and delocalization regimes. Then, we make use of the Webster's transformation [START_REF] Webster | Acoustical impedance, and the theory of horns and of the phonograp[END_REF][START_REF] Martin | On webster's horn equation and some generalizations[END_REF] to symmetrize and convert the Helmholtz equation into a Schrödinger equation with an equivalent random potential that preserves the localized nature of high-energy modes. From that, we build an adapted LL to retrieve information on the original classical modes and we calculate their minimal support. This work highlights limitations of the original LLT for systems exhibiting high frequency localization and enriches it by restoring the LL as a predictive and computationally affordable tool to treat Anderson localization.

We start our analysis by examining the phenomenological differences between localization of quantum and classical waves. We first consider the 1D Schrödinger 3), and bounded by the corresponding LLs u ψ and uϕ (black lines). The purple-shaded areas correspond to the supports S ψ 1 and S ϕ 2 for the modes ψ1 and ϕ2, defined by Eq.( 4). An animation of panels (d,h) is provided in the Supplemental Material [32].

eigenvalue problem of Eq.( 1) with a Bernoulli-type random potential V (x) constructed as follows: a unit-length space is divided into N sites on which V either takes the value V min or V max , with equiprobability. An example of possible realization with 256 sites is shown in Fig. 1(a). We then calculate the first 100 eigenmodes and show a selection of them in Fig. 1(b). Localized modes are expected to decay exponentially as |ψ n (x)| ∼ exp(-x/ξ), with ξ the localization length. In Fig. 1(c) we show the measurement ξ for each mode and averaged over a thousand realizations [31]. First modes are indeed localized and delocalization progressively occurs at higher energies. We then compute the LL u ψ from Eq.( 2) and present it in Fig. 1(d). One can see the effectiveness of the LLT, with the matching between the LL's main peaks and the first localized modes. The mathematical essence of the LLT is encapsulated in the following inequality [START_REF] Filoche | Universal mechanism for Anderson and weak localization[END_REF]:

R ψ n def = |ψ n (x)| ||ψ n (x)|| L ∞ E n ≤ u ψ (x) , (3)
which states that the LL u ψ > 0 acts as a supremum and bounds the eigenmodes, when normalised by their energy. In Fig. 1(d), we show how the LL acts as an upper bound for the normalized modes R ψ n . Moreover, an eigenmode ψ n with energy E n possesses the following approximate support [START_REF] Arnold | Localization of eigenfunctions via an effective potential[END_REF]:

S ψ n def = x ∈ Ω : u ψ (x) ≥ 1 E n . (4)
In other words, the support S ψ n defines where most of a mode's "mass" must sit, as illustrated in Fig. 1(d). As E n increases with n, eigenmodes are allowed to occupy a larger region of space and thus to delocalize.

We now perform the same analysis with a classical system, taking the example of the 1D scalar wave equation

- 1 ρ(x) ∇ • (κ(x)∇ϕ n (x)) = ω 2 n ϕ n (x) , (5)
where ρ(x) and κ(x) can be interpreted as a mass density and bulk modulus in the context of acoustics, and ω n the eigenfrequencies playing the role of an energy. Note that in this form, the operator in the LHS of Eq.( 5) is not self-adjoint but could be rewritten as a generalized eigenvalue problem. We further simplify this problem by considering a single random variable, setting ρ(x) = κ(x) = z(x). Such approximation is known as the Webster's horn equation [START_REF] Webster | Acoustical impedance, and the theory of horns and of the phonograp[END_REF][START_REF] Martin | On webster's horn equation and some generalizations[END_REF] and as we will see later on, can be easily mapped into a Schrödinger equation. The random parameter z(x) is constructed in the same way as for V (x), and can take the values z min and z max , see Fig. 1(e). A selection of eigenmodes for Eq.( 5) is shown in Fig. 1(f) along with the averaged measurement of their localization length ξ . Unlike the Schrödinger system, first modes are fully delocalized and localization progressively occurs at higher frequencies, see Fig. 1(g), until ξ reaches the minimal possible value allowed by the system, that is the size of a single site 1/N . In the Supplementary Material, we show that for a homogeneous 2D problem, the choice of boundary conditions can also lead to a modification of the localization regime [32].

The phenomenological difference between the quantum and classical systems can be intuitively understood from the structure of Eqs.( 1) and [START_REF] Piraud | Anderson localization of matter waves in tailored disordered potentials[END_REF]. We refer to the argument presented in Ref. [START_REF] Van Der Baan | Acoustic wave propagation in one dimensional random media: the wave localization approach[END_REF], which essentially states that for the Schrödinger equation, the potential term is additive to the energy, so that a quantum wave predominantly perceives the effect of the disordered potential at low energies, thus exhibiting low energy mode localization. On the other hand, the classical wave equation can be rewritten in the form of a Laplacian plus another term, equivalent to a potential, but multiplicative to the energy. At low frequencies, a classical wave thus essentially perceives a homogeneous medium, and the first eigenmodes resemble the usual global modes of the domain [START_REF] Fouque | Wave Propagation and Time Reversal in Randomly Layered Media[END_REF], see Fig. 1(f). The random fluctuations become predominant only at higher frequencies, allowing Anderson localization.

We then attempt to compute an equivalent LL for the classical system, for this purpose we solve

- 1 z(x) ∇ • (z(x)∇u ϕ (x)) = 1 . (6)
and show the solution u ϕ in Fig. 1(h) along with a selection of normalized modes R ϕ n , as previously done in Eq.( 3). Despite the supremum R ϕ n ≤ u ϕ being seemingly valid, u ϕ does not here provide any useful information about the mode localization. It is akin to the first eigenmode of the system, and does not possess any peaks or valleys. higher-energy modes, regardless of their localization, rapidly fall orders of magnitude below u ϕ . Equivalently, the support S ϕ n for the first modes already represents a large region of space and quickly converges to the whole domain Ω as the eigenvalue increases, and so as 1/E n decreases. The solution u ϕ thus fails to provide information when the first modes of the system are delocalized.

This naturally brings the need for an alternative version of the LL. To do so, we first use a symmetrized version of the classical wave equation in order to convert it into an equivalent Schrödinger equation. The transformation implies a simple mapping between the two systems as we set ψ n (x) = z(x)ϕ n (x), which leads to

- 1 z(x) ∇ • z(x)∇ ψ n (x) z(x) = E n ψ n (x) . (7)
Expanding the LHS of Eq.( 7), one obtains the Laplacian operator and the equivalent potential

V eq (x) = 1 4 2 z z - |z | 2 z 2 , (8)
see Supplemental Material [32] for details of the transformation and the way we compute V eq (x). Such a transformation is notably known as the Webster's transformation [START_REF] Webster | Acoustical impedance, and the theory of horns and of the phonograp[END_REF][START_REF] Martin | On webster's horn equation and some generalizations[END_REF]. It preserves the energy spectrum and only affects the eigenmodes' shape, here modulated by the factor √ z, see Fig. 2(a) for the first transformed mode. Hence, a localized mode modulated by the delocalized function z(x) remains localized in the transformed basis. This transformation points out a limitation of the LLT, as one can construct, as we did, a random quantum potential leading to high frequency localization and consequently to an uninformative LL.

We thus adapt the LL by adding an energy shift to the operator in Eq.( 7), affecting the eigenvalues but not the eigenstates, and we compute the corresponding LL ūψ (x; E s )

- 1 z(x) ∇ • z(x)∇ ūψ (x) z(x) + E s ūψ (x) = 1 . (9)
Expressing this equation back into the original space through inverse Webster's transform, we obtain the final expression for the adapted LL ūϕ (x; E s ):

-∇ • (z(x)∇ū ϕ (x)) + E s ūϕ (x)z(x) = z(x) . ( 10
)
We therefore derive an adapted supremum for the original classical modes

Rϕ

n def = |ϕ n (x)| ||ϕ n (x)|| L ∞ (E n + E s ) z(x) ≤ ūϕ (x) , (11)
and an adapted expression for their support Sϕ

n def = x ∈ Ω : ūϕ (x) ≥ 1 (E n + E s ) z(x) . (12)
Unlike their previous versions and because of the extra z(x) term, Eqs.( 11) and ( 12) are now based on a local energy criteria to bound a mode or to determine its support. The support for the first transformed mode Sϕ 1 , computed according to Eq.( 12) for a given E s , is shown in Fig. 2(a). Given that all eigenmodes form a basis of orthogonal functions ϕ n |ϕ m = δ nm , the mode ϕ m occupying the domain Ω m is thus orthogonal to all the previous modes ϕ n for n < m, which previously occupied a smaller subdomain Ω n ∈ Ω m . If, at a given energy E m , the support Sϕ m suddenly allows for the occupation of a new subdomain Ω m ⊂ Ω, relatively to the previous eigenmode ϕ m-1 , it is then likely that the mode ϕ m will be localized in the newly allowed subdomain Ω m \ Ω (m-1) . This is indeed a more energetically favorable configuration to satisfy the mode orthogonality condition. This is illustrated in Fig. 2(b,c), where we plot two high frequency localized modes with the components of the inequality in Eq.( 12), defining their support. For the first localized mode at n = 66, the energy level 1/(E 66 + E s ) √ z locally appears below the main peak of the LL ūϕ , thus allowing for the new subdomain Ω 66 \Ω 65 to participate as a support for the mode Rϕ 66 , see redshaded area. As the eigenvalue increases, the same process occurs for the LL's secondary peaks, see Fig. 2(c), generating other high energy Anderson localized modes. This process is perfectly captured by the proposed extension of the LLT. We emphasize that the adapted LL ūϕ can be computed for an arbitrary energy shift E s . Yet, the choice of E s has a strong influence on the LL's final shape. For a small shift (E s → 0), the LL remains uninformative as it is akin to a global structure mode [START_REF] Fouque | Wave Propagation and Time Reversal in Randomly Layered Media[END_REF], as in Fig. 1(h). In the limit of large shifts (E s → +∞) the LL converges to the medium's structure (ū ϕ → 1/ √ z). In the first limit the LL does not possess any peaks or valleys, and in the second limit all peaks or valleys have the same values, which is uninformative in both cases. Only intermediary values of E s lead to a useful landscape with distinct local maxima, as shown in Fig. 2(d) where ūϕ (x; E s ) is plotted for different values of E s .

To finally get rid of the arbitrary choice of E s , we propose a method to compute a minimal support for each mode. We have seen that a mode ϕ n must be mostly located in the support Sϕ n defined by Eq.( 12), and this, for all values of E s . By contraposition, we can state that if, at a given position x, there exists a shift E s for which Eq.( 12) is not satisfied, then x must be excluded from Sϕ,n . So for every point of the domain Ω and at a given energy level E n of interest, one can simply check among a previously calculated set of landscape functions ūϕ (x; E s ), if there is a value E s for which Eq.( 12) is not satisfied. In such case this position is excluded from the support of the mode ϕ n . This procedure amounts in computing the intersection of all mode supports associated with the considered shifts, and it defines the minimal support for the modes Sϕ,min

n = x ∈ Ω : u ϕ (x) ≥ 1 E n , (13)
with

u ϕ (x) = max Es    1 ūϕ (x; E s ) z(x) -E s -1    . (14)
In Fig. The function u ϕ defining the minimal support and the example of the minimal support for the 66 th mode are presented in Fig. 2(f). It clearly shows how low energy modes possess an extended support (purple-shaded areas) and how extra supports (red-shaded area) for the localized mode appear at higher energies. Equations ( 13) and ( 14) arise as a powerful tool to predict the position of high energy localized states as well as their minimal energy.

In conclusion we have shown that, although being mathematically valid for classical waves, the LLT is not appropriate to describe high-energy Anderson localization. Such configuration can also occur for quantum waves with a specific potential, such as the one we derived from a single scalar wave parameter. The inversion of the usual localization/delocalization phenomenology greatly complicates the analysis of localized modes and the prediction of their position and shape. Nonetheless, we have introduced a modified LL, combining a non-uniform excitation of the system and a positive energy shift, which allows to recover information on localized modes. The minimal support for each mode can then be constructed by combining landscape functions computed for different energy shifts. This is of course computationally more expensive than solving a single elliptic problem but remains, in practice, an advantageous alternative to solving the original eigenvalue problem [START_REF] Golub | Matrix Computations[END_REF]. We also emphasize that the acoustic wave equation with κ = ρ could be tackled in a general way as a position-dependent mass Schrödinger equation, and with a transformation involving a change of coordinates [START_REF] Plastino | Supersymmetric approach to quantum systems with position-dependent effective mass[END_REF][START_REF] Pena | Isospectral orthogonal polynomials from the darboux transforms[END_REF].

FIG. 1 .

 1 FIG.1. Comparison of mode localization between a quantum (a-d) and a classical scalar system (e-h). (a,e) Random Bernoullitype quantum potential V (x) and classical parameter z(x) with 256 sites. (b,f) Selection of eigenmodes normalized to their max, for the quantum and classical system, respectively. (c,g) Mode localization length ξ , averaged on 1000 realizations (blue lines) along with some single trajectories. The dash part of the blue curves indicates that the exponential fit is less relevant when modes are delocalized[31]. The dashed-red line set at 1/256 defines the smallest possible localization site, hence the limit value for ξ . (d,h) Same selection of modes R ψ n and R ϕ n , now normalized to their energy, see Eq.(3), and bounded by the corresponding LLs u ψ and uϕ (black lines). The purple-shaded areas correspond to the supports S ψ 1 and S ϕ 2 for the modes ψ1 and ϕ2, defined by Eq.(4). An animation of panels (d,h) is provided in the Supplemental Material[32].

8 FIG. 2 .

 82 FIG. 2. (a) Adapted LL ūϕ(x; Es) (black) computed with Es = 2 × 10 4 , and first eigenmode Rϕ 1 (blue). Purple-shaded areas correspond to the support Sϕ 1 defined in Eq.(12). (b,c) Focus on localized modes Rϕ 66 and Rϕ 70 . The dashed-green line corresponds to the RHS of the inequality in Eq.(13) which locally defines the mode's support. The red-shaded area indicates the extra support obtained from the previous eigenvalue. (d) Evolution of the adapted LL ūϕ (re-scaled) for different values of energy shift Es ∈ {10 3 , 2 × 10 4 , 10 6 } (blue to red). (e) Measure of mode support Σ ϕ n (Es) for the same values of shifts Es as in (c), and measure of the minimum support Σ min n (dashed-green). (f) Function u (x) defining the minimal mode support (black). The purple-shaded area is the support for the 65 th eigenvalue while the red-dashed area indicates the extra support obtained for the 66 th one. An animation of panels (a,f) for a set of modes is provided the Supplemental Material [32].

  2(e) we show the measure of the mode support Σ n (E s ) = Ω Sϕ n (E s )dx for different values of shift, along with the measure of the minimal support Σ min n = Ω Sϕ,min n dx. This highlights the drawback in using a single shift value E s : the landscape ūϕ (x; E s ) and support Sϕ n (E s ) only provide useful information in a certain range of energies (or mode number), close to E s . Above this value, Σ n (E s ) converges to the whole domain Ω, hence being not informative anymore.

Acknowledgments-This research was supported by the Excellence Initiative of Aix-Marseille University -A*Midex, a French "Investissements d'Avenir" program, and the Région Sud (project AndaLoca n°2020 03026).

Supplemental Material : Crossover Between Quantum and Classical Waves and High

Frequency Localization Landscapes

David Colas, 1 Cédric Bellis, 1 Bruno Lombard 1 and Régis Cottereau 1, † 1 Aix Marseille Univ, CNRS, Centrale Marseille, LMA UMR 7031, Marseille, France † Electronic address: cottereau@lma.cnrs-mrs.fr (Dated: April 26, 2022) In this Supplemental Material we provide additional details on the Webster's transformation and the potential function used in the main text, and we present a 2D example showing the effect of the boundary conditions on the mode localization. Finally, we describe the content of the two Supplementary Videos. Equations and figures from the main text are here quoted with numbers, whereas those from the supplemental material are prefixed by "S".

A. Effect of boundary conditions on 2D wave localization

We provide a concrete and illustrative example of the effect of the boundary conditions (BCs) on the mode localization for a homogeneous 2D surface randomly pierced with holes. We always assume homogeneous Dirichlet BCs on the edge of the 2D domain and we either consider homogeneous Dirichlet or homogeneous Neumann BCs on the edges of the holes. Using a finiteelement method, we compute the first modes for the eigenvalue problem -∆ψ n = Eψ n and the associated localization landscapes -∆u ψ = 1, with the corresponding BCs. The localization landscapes along with the two first modes for each case are shown in Fig. S1. The case with homogeneous Dirichlet BCs everywhere is analog to the one presented in [START_REF] Dhillon | Localization of ultrasound in 2d phononic crystal with randomly oriented asymmetric scatterers[END_REF], and exhibits low frequency mode localization, with the main peaks of the localization landscape indicating the position and shape of the first localized eigenmodes, see panels (a,c,e). However, with homogeneous Neumann BCs on the edges of the holes, the first modes are essentially unaffected by the presence of holes and thus they are delocalized. The localization landscape is akin to the first global structure mode, so uninformative, see panels (b,d,f).

B. Webster's transformation

The Webster's transformation maps a one-parameter classical wave equation into a Schrödinger equation (SE). Strictly speaking, the transformation maps their associated spectral problems. Let us start from the later, a time-independent SE:

where E n is the eigenenergy of the associated quantum eigenmode ψ n (x). We set ψ n (x) = y(x)ϕ n (x), with ϕ n (x) the classical modes and where y(x) is to be determined. Substitution into Eq. (S1) yields

To get rid of the linear term in ϕ, we define y(x) as a nontrivial solution of the homogeneous differential equation

This lets us with the equation

Now, setting z(x) = y 2 (x) in the above equation, we get

which can be recast as

that is a one-parameter classical wave equation, and also called Webster's horn equation. Also, given Eq. (S3) and that z(x) = y(x) 2 , we can derive a unique expression for the equivalent Schrödinger potential which reads

Reciprocally, if one wants, from a given potential V (x), to determine the corresponding classical parameter z(x), one needs to solve Eq. (S3) for z(x), which requires two integration constants.

C. Acoustic structure and associated potential

The Bernoulli-type random parameters from the main text essentially consist in a succession of rectangle functions with different widths. Practically, to numerically construct our classical 1D random structure z(x), we choose a smoothed approximation of rectangular functions based on error functions. At a site k, we thus have µ controls the smoothness of the rectangle's walls, assuming 0 < µ σ. In the limit of µ → 0, z k (x) defines a straight rectangle function. This choice is particularly convenient for the derivation and the numerical computation of the Webster's potential V eq (x), see Eq. (S7), since it essentially involves first and second derivatives of z(x). They end up being Gaussian and first derivative of Gaussian functions, instead of Dirac delta functions and their derivatives if straight rectangle functions are chosen. The rectangle (µ → 0) and smoothed rectangle functions (µ > 0) are plotted in Fig. S2(a), along with the first (b) and second derivative (c), for the smoothed case. The potential function from Eq. (S7) is shown in Fig. S2(d). A section of the equivalent potential computed from the main text's parameter z(x) is also shown in Fig. S2(e). Performing the Webster's transform here leads to an equivalent potential which resembles a complicated "'Gaussian" comb.

D. Supplementary Videos

Two videos are provided. Supplementary video S1 corresponds to an animated version of Fig. 1(b,d,f,h) for the first 100 eigenmodes of the quantum (top) and classical (bottom) systems. Normalized modes R ψ,n and R ϕ,n with their respected localization landscapes and supports are shown on the left. Eigenmodes simply normalized to