Marek Hołyński
email: marek.holynski@gmail.com

Early Computer Development in Poland

Keywords:

The paper describes the history of early computing in Poland from the establishment of the Mathematical Apparatuses Group in 1948, which later changed into the independent Mathematical Apparatuses Division of the Polish Academy of Sciences and then finally became the Institute of Mathematical Machines, where the ZAM computers were designed and produced. Two other research and development centers, which were most important in the late 1950s and 1960s are also described: the Faculty of Telecommunications of the Warsaw Institute of Technology (UMC machines) and Elwro (Odra series). When the production of RYAD computers started in 1973, the production of these machines had to be dropped.

Introduction

The early development of computers in Poland has been very sparsely documented in English-language publications. English-language articles published in professional periodicals during that period usually detailed specific technical solutions, and seldom gave sufficient historical context. PIPS -the Polish Information Processing Society (Polskie Towarzystwo Informatyczne -PTI) has been trying to collect all of these articles, but currently only has two of them in its archives [START_REF] Łukaszewicz | On the Beginning of Computer Development in Poland[END_REF][START_REF] Marczyński | The first Seven Years of Polish Digital Computers[END_REF], which might be partially considered historical studies. Both concentrate primarily on the personal achievements of the authors and dedicate less detail to other works carried out in parallel. Moreover, these works focus on the 1940s and '50s, and do not take into account the consequences of these developments in the following decades.

The Polish-language resources are richer [START_REF] Madey | Początki informatyki w Polsce [The beginnings of computer science in Poland[END_REF][START_REF] Targowski | Informatics. Models of development and systems[END_REF][START_REF] Łukaszewicz | O początkach informatyki w Polsce [On the beginnings of computer science in Poland]. Materiały konferencji PTI "40 lat informatyki w Polsce[END_REF][START_REF] Nowakowski | 50 lat polskich komputerów, historia romantyczna[END_REF][START_REF] Miś | To już pół wieku [It's been half a century[END_REF][START_REF] Jasicki | Zastosowanie maszyny matematycznej PARK do obliczania stopnia kompensacji mocy biernej w sieciach elektroenergetycznych [The use of the PARK mathematical machine to calculate the degree of reactive power compensation in power grids[END_REF][START_REF] Hołyński | Raport w sprawie ANOPS-a[END_REF][START_REF] Bilski | Wrocławskie Zakłady Elektroniczne ELWRO[END_REF][START_REF] Maćkowiak | Polskie komputery rodziły się w ELWRO [Polish computers were born in ELWRO[END_REF][START_REF] Lesiński | Komputer Odra 1103 [Computer Odra 1103[END_REF][START_REF] Bilski | Wrocławskie Zakłady Elektroniczne. Okres komputerów Odra 1300 [Wroclaw Electronic Factory. Period of computers Odra 1300[END_REF][START_REF] Kulisiewicz | Własne konstrukcje, licencje, klony [Own constructions[END_REF], but even they do not provide a full picture. Some, which are not referenced here, were simply juxtapositions of dates and milestones. Others were limited to very specific computer applications in particular sectors of the economy or overly personalized memories. Often times, attempts at a more comprehensive summary of the subject contradicted prior work or were continuations of long-outdated controversial discussions.

As the anniversary of the 70th birthday of Polish informatics drew near, PIPS was tasked with the coordination of jubilee ventures prepared by various government institutions, universities, associations and companies. We had to organize our historical knowledge to fill existing gaps and develop a comprehensive description of the history of Polish computing. Not only because this story needs to be better known, but also to serve as a reference during the number of presentations for this event. Moreover, such an overview should not be reduced to a dry litany of technical parameters, people, and inventions. Instead, it should serve as a historical narrative that is interesting even to the ordinary reader or listener. This task was originally assigned to me as chairman of the national commemoration committee.

The preliminary material describing the years 1949-73 consisted of 65 pages. Several corrections were made after consultation with the PIPS's historical section and a number of discussions with the Society board of directors. It was then printed in sections to consecutive PIPS's bulletins [START_REF] Hołyński | lecie polskiej informatyki[END_REF][START_REF] Hołyński | Analog czy cyfra?[END_REF][START_REF] Hołyński | Pierwszy polski komputer[END_REF][START_REF] Hołyński | We have ZAMs[END_REF][START_REF] Hołyński | Wczesne lata 60-te: wysyp maszyn[END_REF][START_REF] Hołyński | Naśladować innych, czy rozwijać własne pomysły? [Imitate others or develop your own ideas?[END_REF], which are regularly distributed in paper and electronically to all members. Their comments and additions have also been taken into account. The following article is a significantly compressed English version of that text.

The initial part of this paper draws partially from two previously cited articles and the missing data is supplemented by information about other parallel achievements. The data presented has been correlated with facts established in the course of my work at the Institute of Mathematical Machines (IMM) from 1970-81. It was then the country's leading R&D institution in this field, and still employed most of the pioneers from early days of Polish computer science. These pioneers were very happy to share their memories of the 1950s and 1960s, and I was equally happy to write them down.

Two other important resources for historical information used in this paper should also be mentioned. In 2008, I became the IMM director, which gave me direct access to the institution's rich archives. In the same year, I was elected as president of the Polish Information Processing Society. Shortly after, we established a historical section in PIPS, which was officially authorized to access collections of state archives. Thus, we began to gather, organize and digitize all available documents and private relations. Most of the material referring to the 1960s and early 1970s comes from these two sources.

Mathematical Apparatuses Group

On December 23, 1948, the weather in Warsaw was particularly bad. Wet snow continued to fall as the inhabitants of the ruined city desperately tried to salvage what they could of their holidays with a meager meal for their family. Only a small group of people seemed untroubled by the worries of the upcoming celebrations. These were attendees of a seminar on electronic calculating machines, listening to a talk given by Kazimierz Kuratowski. Kuratowski was a renowned topologist and director of the Institute of Mathematics in the Polish Academy of Sciences. He had just returned from a series of lectures in the United States, where he was shown ENIAC, the first electronic general-purpose computer, dubbed in the press as "the giant brain." His excitement about the newly built machine sparked great motivation among the listeners and led them to the immediately form a new research team.

The team, officially named the Mathematical Apparatuses Group, began building their own computer, despite having access to very limited resources. The researchers, who wore leaky boots and were barely surviving on the food parcels from post-WWII international relief agencies, did not have access to the proper equipment, parts, or even premises to pursue their endeavors. Moreover, the new American advancements in relevant fields were not often shared with the public, much less other countries, due to their applications in the military. Even the technical details which were released did not often reach Poland, as a result of the Iron Curtain.

The Mathematical Apparatuses Group was provided three rooms at the Institute and for quite a while their work remained only on paper. Their first attempts to deal with real devices did not bring significant results. For each damaged module they repaired, another one broke down, and the process would repeat.

3

Analog or Digital?

Finally, in 1953, they were able to get something working: an analog machine built with vacuum tubes, which was called the Differential Equations Analyzer (ARR). It was able to solve complex differential equations with very high accuracy and was used for a number of practical applications, including the design of turbines and aircrafts. The next machine, the Algebraic Linear Equations Analyzer (ARAL), also proved to be useful. In subsequent versions, ARAL-1, -2 and -3 served to solve systems of equations (in fact, not only linear). Both ARR and ARAL were analog machines designed exclusively for a single task and nothing more. At the time, they seemed like the obvious choice because of the team's considerable experience in analog construction and the low efficiency of vacuum tubes. However, the advent of digital devices had brought about much more versatile machines that could be programmed for various types of calculations and were less susceptible to accumulating errors.

The first attempt to construct a digital machine was made in the years 1953-1955 by the team led by Romuald Marczyński. The Electronic Machine for Automatic Calculations (EMAL) was said to perform 2000 additions or subtractions, 450 multiplications, and 230 divisions per second. The solution which allowed for the "fast" memory of this machine consisted of a number of glass tubes filled with mercury, which could often not be sealed adequately, resulting in a health hazard. According to an anecdote passed from one generation of computer scientists to the next, one of the team members was reminded of a commonly used item made of latex that had the perfect dimensions for sealing the glass tubes. While the saleswomen at the pharmacy were not particularly surprised when he requested one hundred pieces, they were taken aback by the request to invoice these highly personal items to the Polish Academy of Sciences.

Unfortunately, this machine was never fully launched -it was too unreliable. The components available at that time in Poland (lamps, connectors, etc.) were of low quality and caused problems too difficult to overcome. It took significant labor to build individual machine modules, which would cease to function in only two or three days. Repairs were often highly complex and seemingly indefinite, constantly requiring new components.

Newspapers at the time which mentioned these projects usually focused on intellectual challenges and conceptual difficulties in creating an "electronic brain," but seldom focused on the real problems with which the researchers were struggling. How could they make a tangle of cables and electronic components (some of which were remains from the German army during occupation) function properly for a reasonable amount of time? Statistics from these years show that, for these devices, downtime for failures and maintenance significantly exceeded the time of effective operation.

4

From ABC to XYZ

In order to consolidate the existing research and design efforts, the Polish Academy of Sciences established the independent Mathematical Apparatuses Division (Zakład Aparatów Matematycznych -ZAM) in 1957. It was there that in the autumn of 1958, the first Polish electronic digital machine was launched with the name XYZ. It used 400 tubes and 2000 diodes, flip-flops on one triode, had drum memory and punched cards for input and output. It could perform up to 1000 arithmetic operations per second and had an internal binary language with symbolic addressing.

The head of the team, prof. Leon Łukaszewicz, when asked by journalists why the machine was named XYZ, would answer: "Well, the version we started with was called ABC, and then there were others." Łukaszewicz was also the representative of the Polish Academy of Sciences to IFIP from 1961 to 1985 and Vice President of IFIP from 1964 to 1968. At that time, commercial applications for such an efficient machine had only just begun to emerge. In 1960 the first improved version of XYZ, called ZAM-2, was built. It included 600 kilobits of memory, teletype and a paper tape reader, and was suitable for mass production. The software, especially System of Automatic Coding SAKO (often dubbed a Polish Fortran), was one of the major selling points of this machine. For the next three years a series of twelve units were produced, a number which was quite significant for the time. ZAM-2 was successfully used in various institutions for numerical calculations and data processing both at home and abroad, as evidenced by the email (quoted here literally) sent by a German user in March 2017, requesting permission to use pictures of the ZAM-2 on his personal website: Dear Mr. Direktor, More than 50 years ago the ironworks in village Hennigsdorf have acquired a computer ZAM 2 GAMMA of the IMM. 10 years I was responsible for the servicing of the machine. With pleasure I would publish this event on my homepage. In the draught the pictures which come from the education in the IMM are included. If it was right to them, that I the draught sends as a ZIP file to receive her approval for the publication.

Permission, of course, was granted. Clearly, the machine had some fans. The Mathematical Apparatuses Division, which by 1962 was operating out of its own building, began its transformation into the Institute of Mathematical Machines (Instytut Maszyn Matematycznych -IMM), as mentioned in the above email. In order to maintain the tradition, the Institute has existed for years under the old name, despite the burden of facing continuous inquiries (especially from younger generations) about what mathematical machines really are.

The word "computer" was only allowed to appear in the Polish language fairly late -in the mid-1970s. Before, it was routinely replaced by the censors' office with the phrase "electronic calculating machine," a compulsory copy of the Russian term for computers. The part of WIT's strategy of was to create computers dedicated for special tasks. The AMC (Administrative Digital Machine) was to be used for data processing; the GEO had a multiprocessor operating system, Fortran translator, several versions of assembler and geodetic programs sets, and revolutionized land measurements in Poland; the ANOPS series was designed to support biomedical research [START_REF] Hołyński | Raport w sprawie ANOPS-a[END_REF], and many of the 150 machines produced worked in medical centers around the world. In fact, the ANOPS was the first Polish computer ordered by the leading American and Canadian research institutions. The most well-known of the computers created at the Warsaw University of Technology was undoubtedly the UMC series (Universal Digital Machine). The UMC-1 prototype was developed in 1960. It was a machine capable of performing 100 operations per second, equipped with a drum memory capacity of 144 kilobits. The construction was so successful that it was chosen for large-scale serial manufacturing. Its blueprints and documentation was handed to Elwro factory, where industrial production began the next year. By 1964, 25 of these machines were made. UMC-1 was a vacuum tube machine, but later versions, such as the UMC-10 (which replaced it in 1965) used transistors and magnetic-core memory.

The third major research center was the Nuclear Research Institute of the Polish Academy of Sciences. A computational group was organized there for supporting the design of nuclear reactors, later transformed into the Department of Applied Mathematics. In 1956, development of the EMAL machine was moved there and shortly after, the machine was put to use. Known as the EMAL-2, it was used for a number of years and then transferred to the Computational Center of the Polish Academy of Sciences.

ELWRO

The Elwro factory, established in 1959 in the city of Wrocław, was to be the production base of the national electronics industry. However, according to everyone who remembers those times, the real intention from the very beginning was to create a factory producing computers [START_REF] Bilski | Wrocławskie Zakłady Elektroniczne ELWRO[END_REF][START_REF] Maćkowiak | Polskie komputery rodziły się w ELWRO [Polish computers were born in ELWRO[END_REF]. In addition to producing UMC machines, Elwro also had its own plans. A development group was created there, which presented the prototype of the Odra 1001 (Odra is a river over which Wrocław is located). For the first time in Poland, floating-point operations were implemented in hardware, although they didn't perform reliably enough.

Their next attempt, the Odra 1003, was reasonably efficient and suitable for serial production, which started in 1964. It was also much smaller than the previous iterations (as it can be seen on Fig. 10). After that, another successful version, the Odra 1013, was developed, which was two times faster than the previous one (a hundred of these were produced in Elwro). There was also one more after this -the Odra 1103 [START_REF] Lesiński | Komputer Odra 1103 [Computer Odra 1103[END_REF].

The next big leap in quality was with the Odra 1204 project. It was already equipped with an operating system, but still, despite the complete Algol translator, with rather modest software. Without a rich set of programs, one could not take full advantage of the potential of even the most capable computer. In order to create necessary software, Elwro would have to organize a large group of expert programmers and provide them with the necessary time. Instead, the plan was made to use the software of one of the reputable foreign companies and avoid this problem. That is, instead of creating software for an existing machine, build a computer on which existing software would work properly.

First, they had to find a company that would agree to such a deal. IBM, the strongest player in the global market at that time, was not interested. However, others were more willing to consider the offer, as long as their equipment was purchased as part of the package. Ultimately, the British firm ICT (International Computers and Tabulators), which soon became part of ICL (International Computers Limited), was selected.

ICT machines in Poland had already been in use for some time and several programmers and engineers were quite accustomed to them. It was decided that Poland would buy two large ICL 1900 computer. In return, Elwro would receive their documentation and software. In Figure 11, the "men in black" are shown in front of the ICT headquarters, having just signed the agreement in July 1967. This contract, as well as the government instructions for the Polish delegation, were published in [START_REF] Bilski | Wrocławskie Zakłady Elektroniczne. Okres komputerów Odra 1300 [Wroclaw Electronic Factory. Period of computers Odra 1300[END_REF].

The same team that made the Odra 1204 was assigned to this project, and as a result, many of the solutions they used were simply adopted from their experiences with the previous machine. Their task this time was more difficult, because the goal was to achieve a system that was fully compliant with the provided British hardware.

Despite the challenge, the team was ultimately successful. The first prototype of the Odra 1304, produced in early 1970, behaved exactly the same way as the ICL 1904, supporting all the same features: the George operating system (then regarded as one of the best in the world), several programming languages (including Algol, Fortran and Cobol, the most widespread at the time) and a library of over a thousand ready-to-use programs. An extensive set of external devices also functioned as expected. Odra 1304 and its successors, Odra 1305 and Odra 1325, had already started being built on an integrated circuit and were considered the best machines in the Eastern Bloc at the beginning of the seventies. With rich software and a full range of external devices, they became the perfect tools for computerization of many enterprises and institutions. In total, 587 units of the Odra 1300 series were produced. Through their lifetime, they enabled the computerization of entire industries such as construction and railway, in addition to various institutions and universities. Top-secret military applications, which were classified for many years, were later revealed to also disclose other use these devices. In 1999, an anti-aircraft system using a (then 20-year-old) Odra 1325 detected and shot down a presumably radar-invisible F-117 during its military mission in Serbia.

Elwro's strategy brought the desired results, but their success has been controversial, even to this day. Many questioned if it was worth copying other work, even if the source material was deemed the best available. Some insist that even when operating at a strict disadvantage, it is best to further develop original ideas, for the sake of scientific discovery and exploration. While clearly the agreement with ICT enabled the massive leap in overcoming complicated technological issues, others still suggest that solving them independently would have been fruitful in the long run.

RYAD

This dilemma -the choice between replicating proven solutions and own creationlater appeared again on a much larger scale. In January 1967, a committee in the Academy of Sciences of the USSR decided that the efforts of Comecon countries should be consolidated and that all countries should join forces to create a unified system of computers. It was decided to follow the model of the IBM 360, the most widespread machines in the world at that time.

Once this decision was made, it became difficult to do anything else. If a country insisted on following their own designs, they were accepting the possibility of political consequences and economic marginalization. It was therefore recognized by the Polish government that participation in this unified program was in the best interest of the country. This undertaking was officially dubbed the Unified System of Electronic Computers, or RYAD.

In later years, the widely accepted theory for this period is that the Soviets forced Polish authorities into the RYAD project, in the same way Polish industries were forced to sell coal and ships at favorable prices. This would not have been an obvious supposition, because according to some officials, there did not seem to be too interest in about Poland's participation in the program, largely as a result of the contention from the ICL contract.

The distribution of tasks showed the Soviet disregard for Polish participation even more clearly. A medium size machine called the R-30 (EC 1030) was to be designed by the Yerevan institute and built in a factory in Kazan. The IMM was tasked with designing its own version of such a machine. Poland as a whole was also involved in this undertaking, although marginally, as they were only tasked with the production of printers, tape memories and terminals.

Prior to this effort, the IMM had been working on the ZAM machines, and in particular the ZAM-3 and ZAM-21 (launched in 1965). The last of this series, the ZAM-41, was not far from the contemporary notion of a computer. It performed 30,000 fixedpoint operations per second, was equipped with a tape memory, line printer, and other peripherals. Its production started in 1966. Once IMM became involved in R-30, it was forced to drop the ZAM line. The IMM teams traveled back and forth to their counterpart institute in Yerevan, returning with cheaply bought Ararat cognac for their friends. In Warsaw, however, their task was met without much enthusiasm something which had been forced on them. This lack of enthusiasm ultimately led to the initiative being taken over from the IMM by the more dynamic Elwro.

As it turned out, Wrocław did quite well. The version of the R-30 built by the Polish team was smaller (it was located in just one cabinet while the Soviet one occupied three), did not need as much energy, and was more reliable. Yerevan's R-30 could not be launched at the Poznań International Fair in 1972, and the machine which did not work had to be presented.

Above all, the Polish version of the R-30 was significantly faster. At the Brno fair, all the RYAD devices were compared using a test prepared by the Czechoslovak Academy of Sciences, which measured the time it took to calculate a set of one million basic arithmetic operations. The R-20, developed in Bulgaria and Minsk, dealt with this test in 200 seconds. The R-30 from Yerevan needed 70 seconds, and the Elwro computer only 7 seconds. Another surprising result was the German R-40 machine, which was five times bigger, and ran the test in 9 seconds. [START_REF] Kulisiewicz | Własne konstrukcje, licencje, klony [Own constructions[END_REF].

Despite initial allegations of violation of the design rules established for the entire Unified System, the Polish computer received a separate R-32 number and in 1973 and decision was made to manufacture it. Even so, the Russians never bought a single piece. Instead, the Kazan factory which had produced the Yerevan R-30 began to implement the solutions used in R-32 in their own modernized R-33.

The development of the Unified System was continued in the 80s by following the more advanced IBM 370 family. As part of this program called RYAD 2 Elwro manufactured R-34 computers until 1987.

Conclusions

As mentioned at beginning, this paper is an extremely compressed English-language version of the material prepared for celebrations of the 70th birthday of Polish informatics in 2018. The Polish report was aimed at presenting the most complete description of the achievements in the years 1949-73 by organizing known facts and supplementing them with first-hand accounts and a large selection of archived materials. Like many such comprehensive reviews, it inevitably still has some gaps. However, after considerable discussion of this material at PIPS meetings and feedback from all Society members, there is hopefully not too much missing.

The Polish-language study, however extensive, still doesn't address some important issues of described period. Firstly, it does not mention the important discussion about what the ultimate purpose of computer would be. This debate first began in the late sixties, and ultimately concluded in a clear shift from the use of computers for strictly scientific use to their use for new applications in business and administration. The study also does not discuss potential benefits or losses as a result of technology transfer, the degree of Soviet influence in Poland or even the evident camaraderie between scientists from different counties despite ideological barriers.

Unfortunately, none of the PIPS members are historians by trade. Therefore, we don't feel qualified to discuss the broader historical and political context of this period -something which certainly merits more discussion. This paper is only a preliminary attempt to gather relevant facts and present them in a way allowing experts with a proper methodology of historical documentation, who may further organize and analyze them. One collection of such source materials is also readily available. Four years ago, PIPS invited its members to submit their memories, which resulted in a book, from which three essays have been cited here [START_REF] Lesiński | Komputer Odra 1103 [Computer Odra 1103[END_REF][START_REF] Bilski | Wrocławskie Zakłady Elektroniczne. Okres komputerów Odra 1300 [Wroclaw Electronic Factory. Period of computers Odra 1300[END_REF][START_REF] Kulisiewicz | Własne konstrukcje, licencje, klony [Own constructions[END_REF].

Fig. 1 .

 1 Fig. 1. Kazimierz Kuratowski. Source: Polish Wikipedia

Fig. 2 .

 2 Fig. 2. Differential Equations Analyzer. Source: Archives of the Institute of Mathematical Machines with permission of the Institute's director.

Fig. 3 .

 3 Fig. 3. Romuald Marczyński (in a white shirt) while working on the Electronic Machine for Automatic Calculations (EMAL) [2].

Fig. 4 .

 4 Fig. 4. XYZ digital computer. Source: Archives of the Institute of Mathematical Machines with permission of the Institute's director.

Fig. 5 .

 5 Fig. 5. Leon Łukaszewicz

Fig. 6 .

 6 Fig. 6. ZAM-2. Source: Archives of the Institute of Mathematical Machines with permission of the Institute's director.

Fig. 7 .

 7 Fig. 7. Instytut Maszyn Matematycznych. Source: Archives of the Institute of Mathematical Machines with permission of the Institute's director.

Fig. 8 .

 8 Fig. 8. Antoni Kiliński. Source: Polish Wikipedia

Fig. 9 .

 9 Fig. 9. UMC-1 at the exhibition in Moscow; drum memory was exposed in the foreground.Source: Polish Wikipedia

Fig. 10 .

 10 Fig. 10. Launching Odra 1003. Source: Memoirs of Mr. Wojciech Lipko (https://polskiekomputery.pl/mgr-inz-wojciech-lipko-wspomnienia, last accessed 2018/11/02), who is visible in the foreground in a striped shirt.

Fig. 11 .

 11 Fig. 11. Polish delegation in front of the ICT headquarters [13].

Fig. 12 .

 12 Fig. 12. ODRA 1304. Source: Polish Wikipedia

Fig. 13 .

 13 Fig. 13. ZAM-41. Source: Archives of the Institute of Mathematical Machines with permission of the Institute's director.

Fig. 14 .

 14 Fig. 14. R-32. Source: Polish Wikipedia


domain: spi phys

text:
Adrien Mas

Guillaume Druart

Maxime Vaché

Sylvain Favier

Aymeric Alazarine

Eric Compain

Nathalie Morin

Nicolas Guérineau

Nicolas Guerineau

Kaleidoscope-based multi-view infrared system

teaching and research institutions in France or abroad, or from public or private research centers.

Snapshot multi-view systems can manage advanced imagery functions [1]. Thus, they are considered for a wide range of applications such as, polarization, miniaturization, superresolution, 3D imagery and spectral imagery [2][3]. Different optical architectures have been proposed to meet these new imagery needs, exploring multi-aperture, single-aperture or hybrid concepts [4]. To cover a wide range of applications in the infrared spectral bandwidth, we are looking for a design with a good spatial resolution for pixel co-registration and fields of view ranging from 10 to 60 degrees. We intend to use uncooled infrared detectors, especially microbolometers, in the Long Wavelength Infrared (LWIR). The signal delivered by a microbolometer can easily drift with the environmental temperature. Therefore, for remote sensing applications, it is worth producing the images on a single Focal Plane Array (FPA) before processing them so each image drifts in the same way. In this paper, we will thus focus on infrared designs able to produce multiple images on a single FPA without mentioning systems based on array of cameras [1]. Moreover, using a microbolometer implies a high numerical aperture of the optical system to maximize the signal-to-noise ratio, typically F/1. One of the most classic designs for achieving multiple images on a single detector is the TOMBO multi-channel architecture [5] which has been widely studied. This concept uses an array of imaging elements such as a monolithic lenslet array, to juxtapose several optical systems and to produce these multiple images on a single FPA. A LWIR uncooled TOMBO camera, has already been made [6].

However, its major drawback relies on the relationship between the F-number (F#) and the field of view (FOV) to avoid blind areas on the FPA or field overlay between adjacent channels. Indeed, in order to maximize the number of useful pixels associated with a single lenslet of the TOMBO system while having the lowest Fnumber, FOV (in radians) and F# are linked by the following equation [7]:

𝐹𝐹𝐹𝐹𝐹𝐹 = 1 𝐹𝐹# . (1)
Validating Eq. ( 1) means that the lenses have the same size as their associated subimages. If we allow that the size of the lenses can be smaller than their subimages, Eq. ( 1) becomes 𝐹𝐹# ≥ 1/𝐹𝐹𝐹𝐹𝐹𝐹.

Therefore, for a FOV of 10°, the F-number should be greater than 6.

This F-number is not compatible when using a microbolometer. In other words, with the requirement of a F-number of 1, the FOV should be greater than 60° for an optimal use of pixels and a smaller FOV will yield unused pixel areas on the FPA between the subimages, as observed in [6]. To overcome this constraint, Carles's team proposed a two-stage freeform TOMBO architecture allowing an increase of the numerical aperture by using off-axis optics to tilt the optical axis of each channel [8]. Freeform surfaces are used to correct the induced off-axis aberrations generated by this approach. With this design, the unused pixel area can be reduced for small field of view systems (in [8], the field of view is equal to 8°). However, freeform surfaces that can be different from one channel to another increase drastically the complexity of the optical system. Since unconventional lenses have to be made and aligned, this solution becomes difficult to apply for more than nine images. Finally, multichannel architectures must manage crosstalk between the different channels, usually by using walls, since a field stop is not easily implemented for high numerical aperture [9]. However, the grazing incidence of rays on these walls can lead to problematic parasitic reflections. We conclude from the state of the art that multi-aperture concepts have limitations for high aperture systems with narrow FOVs. Light-field type cameras were studied too [10][11]. Among these different concepts, the single aperture light-field design based on a kaleidoscope does not suffer from the limitations of multi-aperture systems [12] and has a field stop that prevents crosstalk. The principle of this optical architecture is to place, between a first imaging lens and a re-imaging one, a kaleidoscope which is a tube made of 4 plane mirrors. A sketch of this design is given in Fig. 1. An intermediate image is formed by the first lens at the entrance of the kaleidoscope. By reflection in the kaleidoscope, the cones of incident beams are divided into N×N cones with different orientations that form virtual copies of the image produced by the first lens at the intermediate image plane. This array of images is then focused by the second lens on the FPA. In [12], a diffuser is used at the entrance of the kaleidoscope to diffuse the cone of incident beams and send it on the mirrors of the kaleidoscope. So a large number of images are formed independently of the aperture of the front lens and off-the-shelf lenses can be used. However, a diffuser induces serious loss of light and image quality and this approach is thus not adapted for systems using an uncooled detector. In [12], authors also proposed a light-field imaging system without diffuser by adding a field lens next to the entrance of the kaleidoscope. As we will see later, if carefully designed, this approach is very interesting for efficient light gathering. However in [12], the use of off-the-shelf lenses makes this approach difficult and the authors use a long kaleidoscope resulting in a decrease of the numerical aperture of the whole system. In this paper we demonstrate that a kaleidoscope concept can be adapted to meet the light efficiency constraints of uncooled infrared detectors. We demonstrate the FOV-F# uncoupling capacity of this system. Finally, we highlight the key dimensions of the architecture to ensure a good distribution of light between the different subimages while maintaining a good compactness thanks to a careful design of all the subsets: the front lens, the kaleidoscope and the re-imaging lens. As a consequence, custom-made optics have to be designed.

Fig. 1. Structure of the kaleidoscope architecture

The kaleidoscope is the central part of the design and impacts the dimensioning of the front lens and of the re-imaging lens for an optimal configuration. The front lens images a scene situated at infinity on an intermediate image plane at the entrance of the kaleidoscope. This way, the entrance section of the kaleidoscope is used as a field stop and its section defines the shape of the images. The presence of a field stop prevents crosstalk between the different subimages. A mask can be placed at the entrance of the kaleidoscope to limit grazing rays on the mirror. However the more the mask overflows on the intermediate image, the more unused pixels will appear between each image on the FPA. At first approximation, the entrance height Hk of the kaleidoscope is linked to the system's field of view FOV (in radians) and to the focal length f1' of the front lens by the following relation:

𝐻𝐻 𝑘𝑘 ≅ 𝑓𝑓 1 ′ 𝐹𝐹𝐹𝐹𝐹𝐹 . (2)
Without a diffuser at the entrance of the kaleidoscope, the front lens needs to have an adequate aperture in order to maximize the light gathering capability. To guarantee a uniform light distribution between the images, the numerical aperture at the intermediate image plane 𝑁𝑁𝑁𝑁 𝐼𝐼𝐼𝐼𝐼𝐼 should be divided inside the kaleidoscope into NxN numerical apertures with similar cone angles as shown Fig. 1.

Then the numerical aperture at the exit of the kaleidoscope NAexit is given by the following equation:

𝑁𝑁𝑁𝑁 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = 𝑁𝑁𝑁𝑁 𝐼𝐼𝐼𝐼𝐼𝐼 𝑁𝑁 .
(

) 3
As a consequence, to have a uniform light distribution between the images, the dimension of the kaleidoscope is linked to the numerical aperture at the intermediate image plane. Fig. 2 illustrates this condition. Rays contributing to a subimage are plotted in a dedicated color: green for the top image, blue for the bottom image and red for the central image. (4) Fig. 2.B satisfies condition (4) and the light distribution is fairly uniform between the image points. The kaleidoscope system was designed to produce 3x3 images. A higher numerical aperture than condition (4) will add rays having double reflection on opposite mirrors and will produce additional unwanted images. It is clear from this figure that the entrance pupil is actually vertically divided into 3 sub-pupils, each associated with a subimage. In this case, the front lens is equivalent to an array of 3x3 imaging apertures. Now we have to guarantee Eq. ( 3) over the field of view. In [13], S. Pacheco et al. optimize a telecentric front lens as illustrated in Fig. 3.A. For simplicity, we represent the front lens and the re-imaging lens as ideal paraxial lenses in ZEMAX software. The configuration of Fig. 3.A has a telecentric front lens and we notice that an extra image point appears, associated with black rays, due to a double reflection inside the kaleidoscope. The upper image in green, obtained through reflection on the bottom mirror, receives less light than its counterpart at the bottom in blue. With the telecentric front lens, the exit pupil is placed at infinity. In fact, to fulfill the condition of Eq. ( 3), the exit pupil of the front lens needs to be at the exit section of the kaleidoscope as illustrated in Fig. 3.B. In Fig. 3.B, we add a field lens in the intermediate image plane to place the exit pupil of the front lens at the exit of the kaleidoscope.

In this case, we clearly see that the light is better distributed between image points and we avoid the double reflections inside the kaleidoscope. This constraint of having an exit pupil after the image plane for an imaging lens is not common and a custom-made front lens has to be designed to guarantee a good image quality. We have therefore defined the design rules for an optimal distribution of light between sub-images. Moreover, this re-imaging lens must have its entrance pupil at the exit of the kaleidoscope and must guarantee the numerical aperture of the whole system 𝑁𝑁𝑁𝑁 𝑠𝑠𝑠𝑠𝑠𝑠𝑒𝑒 which is linked to the numerical aperture at the intermediate image plane and the features of the kaleidoscope. Combining Eq. ( 3) and Eq. ( 5) also gives an important kaleidoscope's feature:

𝑁𝑁𝑁𝑁 𝑠𝑠𝑠𝑠𝑠𝑠𝑒𝑒 = 𝑁𝑁𝑁𝑁 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑀𝑀 = 𝑁𝑁𝑁𝑁 𝐼𝐼𝐼𝐼𝐼𝐼 𝐻𝐻 𝑘𝑘 𝑡𝑡 𝑑𝑑𝑒𝑒𝑒𝑒 . ( 6
)
In our demonstrator, we decide that 𝑡𝑡 𝑑𝑑𝑒𝑒𝑒𝑒 is equal to the height 𝐻𝐻 𝑘𝑘 of the kaleidoscope. In that case, the re-imaging lens divides the focal length of each sub-image by a factor N compared to the front lens focal length (Eq. ( 5)). However, the entrance pupil associated with each subimage is divided by N by the kaleidoscope (Eq. ( 3)). Finally, a pixel would see the same optical etendue in the intermediate image plane as in the FPA and the numerical aperture of each subimage NAsyst is equal to NAIIP as shown in Eq. ( 6). Sometimes, it could be easier to design a front lens with a lower numerical aperture, while keeping the numerical aperture and the FOV of the whole system constant. In that case, Hk is greater than 𝑡𝑡 𝑑𝑑𝑒𝑒𝑒𝑒 and Eq. ( 2) tells us that the focal length of the front lens increases. As a consequence, the size of the whole system increases. Finally, by combining Eq. ( 6) with Eq. ( 4) and Eq. ( 2), by assuming that 𝑡𝑡 𝑑𝑑𝑒𝑒𝑒𝑒 = 𝐻𝐻 𝑘𝑘 and by considering that the F-number of the whole system (F#syst) is linked to NAsyst by the following relation 𝐹𝐹# 𝑠𝑠𝑠𝑠𝑠𝑠𝑒𝑒 = 1/(2 × 𝑁𝑁𝑁𝑁 𝑠𝑠𝑠𝑠𝑠𝑠𝑒𝑒 ), we obtain another feature of this system:

𝐹𝐹# 𝑠𝑠𝑠𝑠𝑠𝑠𝑒𝑒 = 1 𝐹𝐹𝐹𝐹𝐹𝐹 𝐿𝐿 𝑘𝑘 𝑁𝑁𝑓𝑓 1 ′ . (7)
In comparison with Eq. ( 1) that describes an optimal TOMBO design, Eq. ( 7) shows that with a kaleidoscope system, the Fnumber is not directly linked with the FOV, which is an important property of this design. Following the previously detailed design rules, we have designed a kaleidoscope-based optical system with the software ZEMAX. We wanted to limit as much as possible the number of optics and the number of aspheric surfaces to keep a simple manufacturing system. Thus, the system could easily be improved optically, to the detriment of manufacturing simplicity and optics alignment. We converge to the architecture illustrated in Fig. 4. It is opened at F/1 and has a field of view of 30°. The kaleidoscope has a square height of 9mm wide and 25mm long. The re-imaging lens has a magnification close to 1/3 and the multi-imaging system produces 3x3 images. The central image is produced without reflection in the kaleidoscope; vertical and horizontal images are induced by a single reflection on the mirrors and diagonal images are induced by double reflection on orthogonal mirrors. The FPA chosen is an uncooled 640x480 detector with a pixel pitch of 25µm. The total length of the optical system is 110mm for a focal length of 6mm. In Fig. 4., the central field is represented to show the image points created. The theoretical polychromatic (8µm-12µm) Modulation Transfer Functions (MTFs) for different fields of view and for the central sub-image, a side sub-image and a diagonal sub-image are illustrated in Fig. 5. The image quality of the system is satisfactory for the 9 images for a 25μm pixel pitch FPA since the MTFs have contrast up to the Nyquist frequency at 20 cycles/mm. Fig. 6 illustrates the light distribution between the sub-images. We see that it is almost the same between the central image and the side images but it is lower in the diagonal images. Indeed, we have dimensioned the kaleidoscope so that it is as compact as possible while distributing the light as well as possible on the vertical and horizontal images. However, the kaleidoscope is square-sectioned and our system is not optimized for the diagonal sub-images. The solution would be to increase the numerical aperture at the intermediate image plane to increase the luminous flux in diagonal images. Given the relation ( 4), the condition would become:

𝑁𝑁𝑁𝑁 𝐼𝐼𝐼𝐼𝐼𝐼 = 3 sin �atan � √2𝐻𝐻 𝑘𝑘 2𝐿𝐿 𝑘𝑘 �� . (8)
In our case, this increases drastically the numerical aperture and complicates the front objective. Indeed, a F-number equal to 0.75 is very difficult to obtain. We observe extra images resulting from double reflections on opposite mirrors which are due to an entrance pupil not perfectly defined. Thus, the numerical aperture of the rays entering the kaleidoscope is too high to avoid opposite double reflections.

The realization and the characterization of the prototype show that we have been able to adapt the kaleidoscope architecture to an application in the LWIR with an uncooled detector. Compared to a TOMBO design, this multi-imaging system has the following advantages. Firstly, it can answer to the high numerical aperture challenge while exploiting a maximum of pixels on a FPA by decoupling the field of view and the F-number. For this reason, it is particularly suited for systems requiring both a small field of view and a high aperture. Secondly, it has some flexibility to modify the field of view by only changing the front lens. Finally, there is no risk of crosstalk because the entrance of the kaleidoscope acts as a field stop. However, it is less compact than a TOMBO design but we have demonstrated that we can limit the size of the kaleidoscope by a careful design strategy that maximizes the light gathering efficiency. The kaleidoscope-based system uses conventional lenses and plane mirrors and thus it enables rapid and costeffective manufacturing and assembling. Different fields of application based on this kaleidoscope architecture can be explored such as spectral, polarization or 3D imaging.

Fig. 8. On the left, an image of a hand through the kaleidoscope system which clearly shows orientations of each 9 images (framed in red); on the right, an image of a roof where we can distinguish small details like an overhead electrical wire that highlights the image quality of the system. Extra images resulting from double reflections on opposite mirrors are framed in blue.

Fig. 2 .

 2 Fig. 2. Impact of the numerical aperture of the front lens, for the kaleidoscope's given dimension, on the formation of the image points for A) a numerical aperture undersized compared to the kaleidoscope dimension, B) an adapted numerical aperture.

Fig. 2 .

 2 Fig. 2.A shows an undersized numerical aperture of the front lens compared to the kaleidoscope's dimension. Indeed, images induced by a single reflection have less luminous flux than the central image. As a consequence, the numerical aperture on the intermediate image plane of height Hk and the length Lk of the kaleidoscope need to satisfy the following equation: 𝑁𝑁𝑁𝑁 𝐼𝐼𝐼𝐼𝐼𝐼 = 𝑁𝑁 sin �atan � 𝐻𝐻 𝑘𝑘 2𝐿𝐿 𝑘𝑘 �� ≈ 𝑁𝑁 𝐻𝐻 𝑘𝑘 2𝐿𝐿 𝑘𝑘 .

Fig. 3 .

 3 Fig. 3. Impact of the orientation of rays entering the kaleidoscope on the image formation in the field for A) a telecentric front lens, B) an exit pupil of the front lens in the exit plane of the kaleidoscope. The re-imaging lens needs to re-image the array of images, having a height 𝑁𝑁 × 𝐻𝐻 𝑘𝑘 in the intermediate image plane, on the FPA having a height 𝑡𝑡 𝑑𝑑𝑒𝑒𝑒𝑒 . This means that the re-imaging lens must guarantee the following magnification M: 𝑀𝑀 = 𝑡𝑡 𝑑𝑑𝑒𝑒𝑒𝑒 𝑁𝑁𝐻𝐻 𝑘𝑘

Fig. 4 .

 4 Fig. 4. Final infrared architecture based on a kaleidoscope and adapted for a microbolometer. Aspheric surfaces are marked with *.

Fig. 5 .

 5 Fig. 5. Theoretical MTFs for different type of images and for 3 fields of view. On the left, the central image's MTF on the right, a side image's MTF; on the bottom, a diagonal image's MTF.

Fig. 6 .

 6 Fig. 6. Illustration of the distribution of 1 Watt of luminous flux between the different images of a system optimized for 5 images.

Fig. 7 .

 7 Fig. 7. Kaleidoscope prototype.Based on the optical design of Fig.4, a prototype has been made. It is shown in Fig.7. Images are given in Fig.8. We clearly see 3x3 sub-images; a central sub-image and 8 others with orientations that match one or two reflections inside the kaleidoscope. We observe extra images resulting from double reflections on opposite mirrors which are due to an entrance pupil not perfectly defined. Thus, the numerical aperture of the rays entering the kaleidoscope is too high to avoid opposite double reflections.


P A Bushev

J Bourhill

Maxim Goryachev

N Kukharchyk

E Ivanov

Serge Galliou

E Tobar

S Danilishin

M E Tobar

Testing of Generalized Uncertainty Principle With Macroscopic Mechanical Oscillators and Pendulums

HAL is

Introduction

At present, one of the grandest challenges of physics is to unite its two most successful theories: quantum mechanics (QM) and general relativity (GR), into a single unified mathematical framework. Attempting this unification has challenged theorists and mathematicians for several decades and numerous works have highlighted the seeming incompatibility between QM and GR [START_REF] Hossenfelder | Experimental Search for Quantum Gravity[END_REF]. It was generally supposed that this required energies at the Planck scale and therefore beyond the reach of current laboratory technology [START_REF] Amelino-Camelia | [END_REF]. However in the relatively recent publication, I. Pikovsky et al. [3] proposed a new method of testing a set of quantum gravity (QG) theories [4][5][6][7][8] by using ingenuitive interferometric measurement of an optomechanical system. The prediction of most of the QG theories (such as, e.g., string theory) and the physics of black holes lead to the existence of the minimum measurable length set by the Planck length L p = G/c 3 1.6 × 10 -35 m [4,7,8]. This results in the modification of the Heisenberg uncertainty principle (HUP) in such a way as to prohibit the coordinate uncertainty, ∆x ∼ /∆p, from tending to zero as ∆p → ∞ [9][10][11][12][13]. The modified uncertainty relation, known as generalised uncertainty principle (GUP), is model-independent and can be written for a single de-gree of freedom of a quantum system as:

∆x∆p ≥ 2 1 + β 0 ∆p 2 + p 2 M 2 p c 2 , (1)
where β 0 is a dimensionless model parameter, M p = c/G 2.2 × 10 -8 kg is Planck mass and p is the quantum ensemble average of the momentum of the system. The dependence of minimum uncertainty of coordinate on average momentum is questionable but some theories [6,8] explain as reflects the connection of spacetime curvature and the density of energy and matter manifested in Einstein's equations of general relativity.

Other more intuitive form of the GUP, e.g.,

∆x∆p > 2 1 + γ 0 ∆p M p c 2 , (2)
which depends only on the uncertainties of the canonical variables of the particle but not on their mean values is predicted in [7]. To test this theory one need to either measure the deviation of the oscillators groundstate energy E min with respect to its unperturbed value Ω 0 /2 [14], or to test of QG corrections to the dynamics of the quantum uncertainty of the mechanical degree of freedom using pulsed measurement procedure proposed in [3], which requires quantum level of precision.

The lowest modal energies measured in large mechanical systems such as AURIGA detector with effective mass of the mode m eff 1000 kg [14] and in dumbbell sapphire oscillator with m eff 0.3 kg [15] set the limit on the QG model parameter γ 0 3 × 10 33 , which is still too large compared to the predicted values of the order of unity [16].

Theory

From the GUP (1) one can derive the new canonical commutation relation:

[x, p] β0 = i 1 + β 0 p M p c 2 , (3)
that is deformed by the QG correction defined by the model parameter β 0 . As shown by Kempf et al. [6], parameter β 0 defines the scale of the absolutely smallest coordinate uncertainty ∆x min = √ β 0 /(M p c). In this work, we experimentally set an upper limit on the value of the model parameter β 0 using the dynamical implications of the contorted commutator on the oscillations of a high-Q mechanical resonator of mass m and (unperturbed) resonance frequency Ω 0 .

We start our consideration with the simple premise that the modification of the fundamental commutator for a harmonic oscillator is equivalent to the nonlinear modification of the Hamiltonian by means of the perturbative transformation of momentum, p → p -β 0 p3 /(3M 2 p c 2 ), which restores the canonic commutator, [x, p] = i , at the expense of adding the non-linear term to the Hamiltonian of the resonator: Ĥ → Ĥ0 + ∆ Ĥ = p2 /2m + mΩ 2 0 x2 /2 + β 0 p4 / 3m(M p c) 2 . Such non-linear correction results in the dependence of the oscillator resonance frequency on its energy [6,8,17]. The dynamics of the system can be described by a well known Duffing oscillator model characterized by amplitude dependence of the resonance frequency, i.e. so called amplitude-frequency effect [18,19]. The necessary frequency resolution in order to sense subtle QG effects can be estimated by using the following expression:

δΩ(A)/Ω 0 = β 0 m eff Ω 0 A/M p c 2 , (4)
where δΩ = Ω(A) -Ω 0 is the deviation of the amplitudedependent resonance frequency Ω(A) from the unperturbed value Ω 0 , m eff is the effective mass of the mode and A is the oscillation amplitude. So, the experimentally measured dependence of the resonance frequency on the amplitude, particularly its null result, may be used to set an upper limit for the model parameter β 0 . The above mentioned theoretical considerations do not specify, which degree of freedom is subject to the QG corrections. If one considers a center of mass mode, then the scale of perturbation is strongly enhanced for the heavier than the Planck mass oscillators, as compared to individual atoms and molecules in the lattice. For instance, the precise measurement of the Lamb shift in hydrogen yielded an upper bound for the model parameter β 0 < 10 36 [16]. Although, the recent experiments with microscopic high-Q oscillators with effective masses ranging from 10 -11 kg to 10 -5 kg, established the new upper bound for β 0 < 3 × 10 7 [19]. The intrinsic acoustic nonlinearity of micro oscillators prevented to test quantum gravity corrections with the greater precision. In the following we describe an experiments with the sub-kilogram split-bar (SB) sapphire mechanical oscillator, where we demonstrate improvement for the upper value of the correction parameter β 0 compared to the previous work with intermediate range mechanical oscillators [19] by nearly an order of magnitude. In addition to the that, we provide the reasonable estimates of β 0 from experiments with bulk acoustic wave (BAW) quartz resonators yields the limit of 4 × 10 4 . As the consequence of mean value entering in the right-hand side of the Eq.( 1) the systems with higher mass and larger amplitude are preferred. As an example one may take measurements of the period of the physical pendulum in 1936 [START_REF] Atkinson | Proc. Phys. Soc[END_REF], where much lower upper bound of β 0 10 -4 can be established from the deviation of the period dependence of amplitude from the well known Bernoulli non-linearity. However, due to the absence of the evaluation of the pendulum frequency stability the exact upper bound on β 0 can not be obtained.

Measurements of correction strength β0 with sapphire dumbbell oscillator

Microwave oscillators based on electromagnetic Whispering Gallery Mode (WGM) sapphire crystals offer excellent short-and middle-term frequency stability [START_REF] Ivanov | [END_REF] due WGM high quality factors exceeding 10 8 and existence of frequency-temperature turnover points. For these reasons these devices found applications in fundamental tests [22][23][24]. The mechanical modes of sapphire resonators may attain Q M 10 8 -10 9 [START_REF] Braginsky | Systems with small dissipation[END_REF][START_REF] Tobar | [END_REF][27]. The resonance frequencies of WGMs are very sensitive to changes in circumference, height of the cylinder resonator and to strain in the crystal lattice thus yielding the necessary coupling between mechanical and electromagnetic degrees of freedom for the observation of mechanical motion. Yet, no acoustic nonlinearities have been detected for the large sapphire mechanical resonators making these devices an excellent platform for QG tests.

The experimental setup, shown in Fig. 1(a), is based on a cylindrical dumbbell shape or split-bar (SB) sapphire resonator, which is fabricated out of a single crystal HEMEX-grade sapphire fabricated by GT Advanced Technologies Inc., USA. The rotation symmetry axis of the resonator is parallel to the c-axis of the crystal. The SB resonator consists of two bars with diameter 55 mm and height 28 mm, which are separated by the neck of diameter 15 mm and length 8 mm, see Fig. 1(b). Two electromagnetic WGM resonators are formed in each bars and undergo the same mechanical motion, i.e. they oscillate in phase for the breathing mode, which is similar to the fundamental longitudinal mode of the conventional cylindrical resonator of the same length and diameter. The resonance frequency of this mode is Ω 0 /2π = 127.071 kHz and its effective mass is calculated by using finite element modelling m eff = 0.3 kg. In order to maximize mechanical Q-factor, the resonator is suspended via a niobium wire-loop around the neck. The whole construction is placed inside temperature stabilized vacuum chamber at 300 K. The vacuum chamber in turn is placed on vibration isolation platform and kept at a pressure of ∼ 10 -2 mBar.

A parametric transducer is used to detect the mechanical vibrations of the SB resonator, see ref. [15] for the details. For that purpose, the WGM sapphire resonator serves as a dispersive element inside a closed electronic loop, which together with an amplifier and a phase shifter constitute a microwave oscillator operating at the resonance frequency of the chosen WGM mode [START_REF] Leeson | Proc. IEEE[END_REF]. An interferometric frequency control system (FCS) suppresses spurious phase fluctuations and locks the microwave oscillator to the frequency of the WGE 15,1,1 mode at ω WGE /2π 9.774 GHz [START_REF] Ivanov | [END_REF]. The in-loop voltage-controlled attenuator α is used for the parametric excitation of the mechanical vibrations at 127 kHz. Approximately a half of the generated power inside the microwave sapphire oscillator is diverted to the interferometric frequency discriminator (IFD). The output signal of IFD is a linear function of its input frequency and is measured with HP 89410A spectrum analyzer. All instruments are time referenced to the hydrogen maser frequency standard VCH-103. The spatial overlap between microwave and mechanical modes results in the interaction between these degrees of freedom, which can be described by the standard opto-mechanical Hamiltonian Ĥint = -g 0 â † âx, where g 0 is a single photon opto-mechanical coupling, â † , â are raising and lowering operators for the WGM and x is canonical position operator for the center of mass mechanical motion [30]. The microwave signal modulated at the resonance frequency of mechanical mode Ω 0 induces radiation-pressure force which drives mechanical vibrations. The calibration of amplitude of center of mass motion is made by using the standard expression

δu(Ω) = δx(Ω)(du/df )(df /dx), (5)
where df /dx is determined from the amplitude of the output IFD signal δu(Ω). That signal is proportional to the applied modulated power δP

δu(Ω) = χ du df df dx 2 δP, ( 6
)
where χ is the constant describing electromagnetic coupling of the signal and mechanical property of the oscillator [15]. The transduction constant is calculated to be δx/δu = 526 nm/mV. The mechanical response of the SB-resonator to the acoustic excitation in the vicinity of the resonance frequency is shown in Fig. 2(a). The applied excitation signal at 127 kHz is relatively weak resulting in the maximal amplitude of mechanical vibrations of 6 pm. The output signal is measured by using phase-sensitive interferometric setup which results in superposition of dispersive and absorptive quadrature components. The solid curve displays the fit of the experimental data to such composite absorptive-dispersive response and yields the resonance frequency of the mechanical resonator Ω 0 /2π = 127070.9695 ± 0.0003 Hz, its FWHM linewidth Γ M /2π = 3.5 mHz and the 55 • degrees mismatch between the arms of the IFD.

The ringdown measurements of the mechanical vibrations are made in two steps. In the first step the resonance frequency of mechanical vibrations Ω 0 /2π is determined. For that purpose, the radiation pressure force is applied to the resonator for the time sufficient to settle the mechanical vibrations (several minutes). Then, the output signal δu(Ω) is measured for every frequency point in the scanning range of 1 Hz. The resonance frequency corresponds to the point which yields the maximal IFD response δu(Ω). This procedure is repeated for the different excitation amplitudes (20-35 pm) in every single experimental run and detected no resonance frequency shift within accuracy of 10 mHz determined by the resolution bandwidth of FFT analyzer. In the second step, after the mechanical resonance frequency is located, the AM-excitation is turned off, and then the mechanical vibrations are measured as they decrease due to acoustic losses. The amplitude and frequency of the decaying vibrations, i.e. the amplitude and the frequency of the spectral peak, is then tracked and recorded every 0.2 s. For that purpose a marker is placed on the maximum voltage value in the spectra, and its frequency and amplitude is recorded for every time bin. The frequency accuracy of such measurements is determined by the resolution bandwidth of the FFT analyzer, which is set to 5 Hz.

The typical ringdown measurement is presented in Fig. 2(b). For this particular example, the resonant frequency is Ω 0 /2π = 127070.97 Hz. The solid curve shows the fit of the experimental data to the exponential decay with characteristic time constant τ = 173 sec which yields the mechanical quality factor Q M = Ω 0 τ /2 = The open circles correspond to β0 reported in ref. [19]. The closed circle is the estimated β0 for the quartz BAW at LDcut, see ref. [18]. The square is the upper limit for β0 obtained with sapphire split-bar resonator. The triangle shows the estimate of the correction strength from the measurements of the period of the physical pendulum, see ref. [START_REF] Atkinson | Proc. Phys. Soc[END_REF].

3.4 • 10 7 and the same FWHM linewdith Γ M which is found in mechanical response measurements. In addition to the extracting of the parameters of the exponential decay, the Duffing equation was numerically solved in order to attain the best fit parameters for the ringdown amplitudes. Following this procedure we extract the upper limit for the QG model parameter to be β 0 < 6 × 10 11 .

The frequency measurements yield much more stringent limit on β 0 . In all measured ringdown series, there is no evidence of any detectable frequency shift up to the maximum amplitude of mechanical displacement of 75 pm. The null-frequency shift measured in the experiment corresponds to the accuracy of δΩ/Ω 0 = 3.9 × 10 -5 and accordingly to the Eq. 4 yields the upper limit for the QG model parameter β 0 < 5.2 × 10 6 .

The sapphire SB resonator demonstrate a large potential for even more stringent test of β 0 1. Here, we propose two possible ways to improve the experiment. Firstly, the mechanical response (Fig. 2(a)) could be measured for much larger input power δP . It is possible to excite vibrations in sapphire resonator with amplitude of several nanometers [27]. That would result is much higher signal-to-noise ratio and as a consequence would improve the accuracy of determination of the mechanical resonance frequency Ω 0 to be better than 0.1 mHz. Together with an increase of the oscillation amplitude by two-three orders of magnitude that may result at least in 10 8 fold improvement for the upper limit on β 0 . Secondly, one can implement an electromechanical sapphire oscillator by closing a feedback loop with the IFD output signal δu(Ω M ). In that case the uncertainty in determination of frequency shift will be decreased with the integration time T as 1/

√ Ω 0 T . Assuming the driving amplitude of SB resonator of A 1 nm and average time of T = 1 hour, the testable limit β 0 1 is within experimental accuracy.

Estimation of the correction strength with BAW Another mechanical system, namely quartz bulk BAW resonator also constitutes a fruitful platform for precise tests of quantum gravity. This system exhibits high resonance frequency Ω 0 /2π 10 MHz, milligram scale of the effective mass of oscillating modes [31], large Q-factor close to 10 10 at low temperatures [32] and high frequency stability of electromechanical oscillators reaching the level of 5 × 10 -14 . The above listed features of quartz BAW are very attractive for fundamental tests such as Lorentz symmetry [33]. However, we note that quartz crystals possess its own quite strong elastic non-linearities that can mimic the quantum gravity effect. These non-linearities lead to a similar frequency shift, quadratic in amplitude and known as amplitudefrequency effect or isochronism, see ref. [START_REF] Manson | Physical Acoustics: principles and methods[END_REF], p. 245. This effect can be made to nearly vanish by means of an optimal choice of the cut angle of the crystal, known as LD-cut [18,[START_REF] Galliou | [END_REF]. The QG correction strength can be estimated from Eq. 4 and by using the experimental parameters m eff = 5 mg, Ω 0 /2π = 10 MHz, δΩ/2π 1 mHz, A 1 nm. Our estimation yields β 0 4 × 10 4 , which is still limited by elastic non-linearity. In order to single out quantum gravity frequency from such non-linearity, the amplitude frequency shift shall be measured in dependence on the effective mass of the resonating mode. We also believe that experimenting with kilogram scale quartz BAW [36] will result in much more stringent test of the quantum gravity model parameter in regime β 0 1, because of weaker non-linearity due to the lower acoustic energy density and much larger effective mass.

Estimation of the correction strength with physical pendulums

At the present time, the most stringent limit on correction strength β 0 can be by using the data set gathered during experiments with mechanical pendulums. Given relatively large amplitude ∼ 1 cm, large mass of the pendulum ∼ 1 kg and reasonable fractional frequency stability ∼ 10 -7 such system, in principle, seems to be ideal for the testing of generalized commutator described by Eq. [START_REF] Hossenfelder | Experimental Search for Quantum Gravity[END_REF]. The pendulum possesses an intrinsic softening non-linearity which can be calculated exactly. The QG correction is assumed to contribute to the non-linearity of the system and the Eq.( 4) can be written in the following form:

δT (θ) T 0 = -β 0 m2πLθ M p cT 0 2 + 1 16 θ 2 + 11 3072 θ 4 , (7)
where δT (θ) is amplitude dependent deviation of the period of the pendulum T 0 , m is the mass of the pendulum, θ is the angular amplitude of a pendulum and L is its length. The last two terms describes the intrinsic non-linearity of the mathematical pendulum. The dependence between the rate and arc (θ) for the free pendulum was measured already in 1936 by using physical pendulum [START_REF] Atkinson | Proc. Phys. Soc[END_REF] with the length L = 1 m and the mass m 6 kg. By taking this data set, we estimate the β 0 10 -4 . Other experiments with different kind of pendulums carried at different times shows no evidence of the deviation of the oscillation period from the conventional theory of mathematical pendulums [37][38][39], and result in similar order of magnitude for the upper limit for β 0 . However, absence of the knowledge of the important experimental details such as Allan deviation, i.e. frequency stability, the absence of any information on systematic and statistical errors lead us to the qualitative conclusion only that correction strength β 0 1, thus putting quite severe constraints on the GUP described by Eq. 3. For the quantitative measurements of β 0 one has to repeat measurements with pendulums or other systems (SB sapphire resonators or BAWs).

Conclusion

To conclude, we have presented measurement of the upper limit on QG correction strength by using ultrahigh-Q mechanical sapphire resonator with sub-kilogram mass of the resonating mode. In the original work [3], a light pulse is proposed to reflect off an oscillator four times, separated by one quarter the oscillation periodbefore having the its phase measured. Our analysis and experiment shows, that one can attain the same goal in continuous RF measurement, which makes the experiment much simpler and reliable because the oscillation frequency can be measured with more precision compared to any other physical parameter. The overview of results of testing β 0 with mechanical oscillators is presented in Fig. 3, which shows the measured constraints on the QG correction strength in dependence of the effective mass of the mechanical resonator. The heavier oscillators allows for the better determination of the correction strength. In principle, old data sets gathered with mechanical pendulums can be used for the immediate estimation of correction strength which yield qualitative results that β 0 1 and put severe constraints on GUP relation described by Eq.(1). However, the remarkable high-Q and frequency stability of state of the art quartz BAW resonators and SB sapphire resonator in conjunction with low acoustic non-linearities have a great poten-tial for its further applications in precise tests of minimal length scale scenarios for the quantum gravity theories [40], where the ultimate limit on β 0 can be measured very precisely.

The great advantage of the electromechanical (sapphire split-bar) or opto-mechanical systems is that potentially they offer the feasible path towards quantumlimited experiments.

In this case, other forms of GUP(described by Eq.( 2)), which solely depend on quantum-mechanical momentum uncertainty ∆p and have no boost from the momentum amplitude, can be tested. In that respect, the perspectives of utilizing of low-frequency (< 1 Hz) mechanical oscillators remains unclear compared to the high-frequency system (kHz-GHz), where nearly quantum regime [41,42] or even quantum limit [43][44][45] can already be accessed.

The research was supported by the Australian Research Council Centre of Excellence for Engineered Quantum Systems CE170100009. PB thanks R. Blatt, F. Scardigli, M. Plenio, A. Vikman and P. Bosso for valuable discussions. SD would like to thank Lower Saxonian Ministry of Science and Culture that supported his research within the frame of the program Research Line (Forschungslinie) QUANOMET Quantum-and Nano-Metrology. The authors are also very thankful to Y. Chen and the members of the MQM discussion group for insightful conversations that inspired this work.

FIG. 1 :

 1 FIG. 1: (Color online) (a) Simplified experiment schematic. See the text for details. (b) Picture of the sapphire SB resonator. The ruler shows the scale of the system.

FIG. 2 :

 2 FIG. 2: (Color online) (a) Mechanical response of the SB resonator to the acoustic excitation of the in-phase breathing mode around 127 kHz. The solid curve shows the fit to the quadrature signal due to double-balanced mixing. (b) Typical ringdown measurement of the in-phase breathing mode. The solid curve shows the fit to the exponential decay.

FIG. 3 :

 3 FIG. 3: (Color online) The correction strength β0 versus mass of mechanical oscillator determined in various experiments.The open circles correspond to β0 reported in ref.[19]. The closed circle is the estimated β0 for the quartz BAW at LDcut, see ref.[18]. The square is the upper limit for β0 obtained with sapphire split-bar resonator. The triangle shows the estimate of the correction strength from the measurements of the period of the physical pendulum, see ref.[START_REF] Atkinson | Proc. Phys. Soc[END_REF].