 Okay, so hi everybody. You tired? A little bit. I feel low energy in the room. Everybody's just like, ah, I just ate. Listening about, I'm so electronic, I don't know what. Okay, I'll try to make it light. Not too many equations. Just explaining concepts. And of course, for those of you who want to know more or understand more, you know there is a session where you can put your hands up. Okay, so my name is Yan Chembo and I'm working in CNRS, which is the French Council for Research. And I actually work in this building. It's in Besançon in France, in the east of France, very close to Switzerland. And the Femto Institute is an institute where actually we do applied science, applied physics. Applied science, I would say, because we also have computer engineering. So we do research on photonics. That's where I work. Electronics, energy, integrated systems, name it. So of course, I'm very happy to be here for a lot of reasons. The first reason is the first time I stayed here, that was 14 years ago. As a PhD student, there was a school on special temporal chaos. So in O2, I was younger, all my hair was black, slimmer, more energy, more everything. And it was my first time to come here. And it was actually for me a very important meeting because it's when I came here that I met someone that says, hey, you know what, come and do your PhD with me in Spain. So always take the opportunity when you come to a conference to talk to people. That's lesson number one. So I'm happy to be here because it's ICTT, but I'm also very happy to be here because it's Hanson. Some of you might not know, but this conference at the beginning was itinerant. They were going in a lot of southern countries. They've been to Brazil, to India, to China. And in 2010, they went to Cameroon. So at that time, I was already working in France, but since I'm from there, the faculty invited me to help run a session still on optoelectronic systems. So you can recognize some people there. I think Brian is here. I can see Bruce here. Of course, Ari is here. I am there. I was there. And there's this girl here. She came with Brian Hunt to run the session on MATLAB. So we started to talk about physics and math. I realized real quickly that actually, we agreed on a lot of things. So we moved forward under collaboration, talking about physics and math and everything. Thank you. And this is our first publication. I'm not sure we'll get a lot of citations out of it, but definitely, as far as I'm concerned, I'm very proud of this work. But anyway, Mike said, sorry, two days ago that something important in this conference is that it's turned towards you. It's not about us telling you how you have to think, what you have to do. It's about helping you develop your own potential. So interact not necessarily to this extent. Don't get me wrong. But interactions are very, very important amongst yourselves. So now let's talk about physics. We'll talk about complex systems and chaos theory. We're going to talk about delayed systems. I'll explain you exactly what a delayed system is. We're going to talk about chaos cryptography, neuromorphic computing, which are applications of delayed systems, ultra stable micro generation, and whispering gala mode resonators, and we'll have the conclusion that we'll be tomorrow morning. So, oh, you're ready. When I say like, okay, we're going to finish tomorrow morning, you're like, okay, fine, we're here, that's good. So complex system and chaos theory. First thing, it might be a very trivial separation, but I would like to say a word about it. The separation between is linear and non-linear system. Of course, a lot of you here are very familiar about that, but still I want to say a word about it. So here, for example, you have a typical linear system. All of us studied this in high school, a spring that is forced. And of course, since this system is linear, I wonder if the characteristic of linear systems is that the sum of two solution is also a solution. Generally always, you can find analytical solutions when you have linear equations. And the solutions can be complicated. For example, if you have a lot of linear systems coupled, the solution can be complicated, but it will never be complex, good. Now, if you have a non-linear system here, we just added, for example, what we call a dufing term. It means that the spring, instead of having a constant coefficient, has a coefficient that varies with the elongation. Then you have a non-linear system. Generally, a non-linear system, if you have one solution and another, if you sum them, it's not a solution anymore. Generally, you cannot find exact analytical solutions of non-linear systems. And sometimes the solutions can be complex. So that is the fundamental difference between linear and non-linear systems. Good. One of the books that I encourage you to read, I read that book when I was ending Bachelor's Degree and this book has been powered in my life because when I read that book, I decided that this is what I wanted to do. It's James Gleck, the book from James Gleck, Chaos. He talks about complexity, the emergence of chaos theory, the pioneering people that worked into it, into that theory. Of course, Ari Sweeney is in that book. It's too modest of a man to talk about it, but since I am not in the book, I can talk about it. So the book more or less presents the fundamental work of people like Edward Lawrence, who was a meteorologist. And in 63, he wrote this paper that widely went unnoticed. Deterministic non-periodic flow. And he present a very small, very simple system of three equations that actually was deterministic, but still non-periodic. And actually introduced the concept of chaotic systems, but then that's not the way they called it. But basically when we talk about a chaotic system, we are talking about a system that is deterministic. So we're not talking about noise. We talk about something that is deterministic. Unpredictable, so in the long run, you will not be able actually to know what's going on in the system. And sensitive to initial conditions. So I want to say a few words about this sensitivity to initial conditions that is generally called butterfly effect. So this is a talk that Edward Lawrence gave. It was in 72 at MIT. And the title was predictability. Does the flap of a butterfly swing in Brazil set off to another Texas? More or less what he wanted to explain to people is that when you have a complex system, if you change a little thing, in one part of the system actually can have huge consequences later. And this term, butterfly effect, so a butterfly flapping its wings somewhere, creating a tornado elsewhere, entered into a common language. For example, this is a very famous program in France. The title is Le fait papillon, which literally means the butterfly effect. And it is a program that analyzes politics every week all around the world about how people doing small things can have a big effect. I can take a very simple example that all of us know, the Arab Springs. He's just a young guy in Tunisia. He's selling fruits. And one day, a policeman frustrates him. And he gets angry and sets fire on himself. Every day in the Arab world, we have thousands of policemen frustrating thousands of people. But for whatever reason, that day, that guy, he changed everything for hundreds of millions of people. This is exactly what the butterfly effect is. It also went to Hollywood. This is a movie that was released in 2004. I didn't see the movie. And the title is the butterfly effect. And what is written here is like, change one thing, change everything. So of course, you get the idea. I don't know what happened in that movie, but very clearly it was some kind of butterfly effect. There is, I am a big fan of The Simpsons. And there is a very nice episode of The Simpsons, Treehouse of Aurora 5, where if I remember well, someone convinces Homer Simpson that if he can come back to the past and change a little thing, then his moronic life in the future will be different. And of course, he goes to the past, changes small things, but in the future, yes, there is a change, but it's generally not the change that he wants. So now we're going to see how we can implement these concepts of non-linear systems into delayed systems. And of course, I first have to tell you what is a delayed system. Take a very simple example. For example, what we call an inverted pendulum. You take a stick like this and you want to control it. Generally, this is something that you will be able to do. Why? Because the timescale at which the pendulum falls is slow enough for your brain to think exactly about what you want to do. For example, if it falls this way, then you will swing your arm that way. Everything is slow enough for your brain to process it and for your arm to actually implement the control action that your brain commands. But actually if you take a stick that is smaller, like for example a pencil, it will be easy for you to realize that you cannot control the pencil. Why? Because actually the pencil falls actually very fast and your brain doesn't have the time to implement the solution, the control solution to stabilize the pen. But actually you have a delayed system in a lot of configurations, just to name a few satellite and rocket control. In the early 60s, end of the 50s, early 60s, Americans and Russians are racing in the space and they realize that when they send a spacecraft far away, when they send a command signal, actually you need some time for the signal to move from the ground station to the spacecraft. And this has an influence. So they tried to open a mathematics book to know exactly how to solve these problems and there was none. Because mathematicians didn't study that problem. And then that's how you have mathematicians starting to work on delay differential equations. How do you put delay in equations and solve the problems that are modeled by that kind of equations? Morning shower problem. Very probably all of us have this problem in the morning. You want some water that is not too cold, not too hot. So what you do is you switch on first the cold water. That's fine, you know it's cold, no surprise. And then you want to switch on the hot water. The water curse is that of course there is a delay so you're in passion. It doesn't come so fast enough so you just turn the knob at the bottom and when the water comes then it's too hot and then you have to come back. So generally you need like 10 seconds of oscillating before actually finding the temperature you want. This is due to delay. If you didn't have any delay, it will be the easiest thing on the earth to have water that is at the right temperature to take a shower in the morning. Cat traffic modeling complex systems, namely there are zillions of situations where actually delay is important and has to be accounted for. So here I would like to talk about a pathologic case of delay control. And this is mass exploration. So this is a picture that I took. These are the three generations of mass rovers, mass exploration rovers. So here you have Sojourner, it looks like a shoebox that was launched in 1997. You have Spirit and Opportunity that are decides. They were launched in 2004. And here you have Curiosity that was launched in 2012 or maybe 2011, I think it was launched in 2011, but landed in 2012. And which is the size of a car, size of a small car. And what am I talking about this is, first thing I would like to introduce the robots that I'm going to talk about one minute later, Spirit and Opportunity. So these are, I don't even know if I can call them mid-sized. They have the size of something like this, the height. And I don't know, two meters long, something like that. So basically these two robots, so the twin robot, so NASA sent two robots actually to Mars. So Spirit and Opportunity, but they were visually the same robots, so two of them just launched them. And they landed in Mars in 2004. And these rovers were supposed to operate for three months and do 0.6 kilometers. This is like a third of a mile. Well, Spirit, so the first robot explored Mars during five years and did something like 7.7 kilometers, like five miles, instead of the third of a mile that was scheduled initially. And Opportunity has already covered more than 42 kilometers. It's the very first man-made device that rents a marathon outside of the planet Earth. And it is still active. So it's been 12 years that these rovers are out there and still working while actually it was scheduled to be only three months old. So what am I talking about this is that Spirit, the one who only worked for five years, had a problem in 09. What was the problem of Spirit? The problem of Spirit was this. So this is the rear camera of the rover, and these are the front wheels, or maybe the back wheels, I'm not sure. But this is the wheel that is stuck. And now imagine the situation. Most of you here know how to drive. Imagine that you're driving your car. Of course, when you turn right, the car turns right. When you turn left, it turns left. Imagine that if when you turn right, actually the car turns right, but after 10 seconds. Imagine what would be the driving situation in the world when whenever you blink or turn or change gears, the car does that, but 10 seconds later. I think that the driving situation would be amazing. Well, here you don't have 10 seconds. Put that to minimum three, maximum 10 minutes because Mars is from three to 12 minutes light away from us. So when you send a signal to Mars, it takes an average, let's say eight minutes to arrive over there. So of course, there is a guy at JPL at NASA that has a joystick to drive this, but he cannot drive this in real time because there is this delay that is insanely huge and he just cannot do it. So actually what they do is that they have a batch of commands and they just send it as a batch and then the rover does that and after that the rover sends back images and they evaluate if it was good or not. So if of course you drive a car and your car is stuck, you know what to do. But when you have so much delay, then you have a very, very, very complex problem. So just for your information, this is how they solved it. So this is a picture at JPL, so more or less they take the rover, incline it with some angle. Why? Because the gravity in Mars is 40% gravity of Earth. So just to have that, the gravity right, so they have an inclined plane, so everybody here knows how that works. And they were trying to find the strategy for to take the rover out of the sand. At the end of the day, they didn't work, even though it was not totally a failure because since the wheel was stuck and it was dragging, actually it was digging and the NASA scientists could see that there was ice below the sand. So at the end of the day they were like, maybe this was a good thing. And this is a picture that I took when I was there. I was just a postdoc working on clocks, microwaves, so different thing, but on campus over there the coolest guys are the guys working on the robots on Mars. These are the coolest guys on campus, so all of us we just go there and take pictures. And so here that's exactly the moment where they were doing all these tests and if you take care, if you pay attention to this logo actually, one day I arrived on campus and everybody has these t-shirts, free spirit and I was like, what is this? I thought it was some philosophical thing, free spirit, whatever. And I did not understand exactly what it was until I went there and was like, okay, I understand. People on campus were just like, having these t-shirts to encourage the roboticians to find a solution to free spirit for myself. But before dying, spirit gave us this beautiful image. This is one of the best pictures I prefer after the pictures of my son, of course. But here this is a earth selfie. So this is a picture taken by spirit in Mars, taking a picture of us, earth. So we are here. We are barely a pixel. When you move away just one planet, we are barely a pixel. So sometimes when you come on TV in the evening, you switch on, you see that people are fighting, Republicans versus Democrats, whites and blacks and Muslims and Christians and yellow and blue and this. You see people, you see mankind fighting so much and in reality, all of us, we are barely a shiny spot when you just move one planet away. Imagine if you move one solar system away, imagine when you move a galaxy away. So I hope that more people have this image in mind generally when they want to fight each other in this planet. So, delay. Delay also has beyond being a problem when you want to drive a rover on Mars. Actually you have people that are evolutionary biologists that actually study the influence of delay in human evolution because we are an inverted pendulum. I mean, this is an unstable position. The only reason why actually, when you have a kid, it takes him one year, one year and a half to understand exactly how to control this inverted pendulum. And the reason I can control it is that if I'm falling, once again, it's slow enough so that my brain knows exactly how I can do and I have enough strength in my legs to control that. So our height is not something that is random. It's correlated to our brain power and to our strength. So there's no coincidence if you don't find small animals that are distal and that are two feet walking like us. It's just that they will have a brain so big that actually they will fall all the time. Good, so now we are going to talk about optoelectronic oscillators and how the delay in these oscillators can actually help us for a lot of applications. So basically what is an optoelectronic oscillator? It's fairly simple. You just have here a system where you have, in one branch, it's a loop. And in one branch, you have energy flowing in optical form. Generally it's going to be a laser, laser light. Here you have some optical to electric conversion. Energy here will flow in the electric form and here you will have the reverse conversion so from electric to optical. And then you have energy that is just flowing like this being optical and after that electrical, optical, electrical and vice versa. Good. Now where's the time delay here? It's just the round trip time. And now with this simple figure actually encompasses a very large variety of systems in engineering, in electrical and photonic engineering and I will just review some of them very, very quickly. So first thing, I need to show you how our equations look like. I will not dig into that, I will not get into details but at least you need to see how the equations look like in the equation that accounts for delay. This is called an Ikeda equation. Ikeda was a Japanese physicist. He published a seminar paper in 79 I think where he proposed this kind of systems including delay in optical systems and basically you have a variable here. It doesn't matter what it is. In the case of Ikeda originally it was the phase of a laser field. So you have this variable here, a derivative and here you have no linear feedback, no linear because you have cosine square and sorry, here the feedback depends on the phase at a value t minus capital T. So this is the delay. So this is for example a typical paradigmatical delay equation. And here actually this is an engineering view. This system has a low pass filter. It means that all the frequencies that are lower than a cut of frequency actually can be excited in the system. The kind of systems that we used, we refer them as generalized Ikeda equations for two reasons. The first reason is that here instead of having a low pass filter we have a band pass filter. So we have a high and low cut of frequency. So instead of having just a derivative we have a derivative and an integral. So it's like an RLC circuit. The gain is the same but here also the no linear function can be something else than a sinusoidal. It can be an elbow function. It can be whatever you want. So that's why we call these generalized Ikeda equations. And most of the system we study are modeled by this kind of equations. Here you have a typical architecture. I'll not get into the details of it for those who, when you will come to the hands on station you will see that. But in short, here you have lasers, a modulator, fiber delay line. The light is detected and you feed it back. So here you have the feedback loop. Here it's going to be optical form. And here it's going to be electric form. And here you have a derivation to monitor what is going on in the system. I would like to make you listen to a root to chaos, temporal chaos. So here we have an Ikeda system and listen to it. So here we increase the gain. And you see here, oh, sorry. Wow, OK. I'm the only one seeing the video. That is unfortunate. Mark, do you think that you know how to switch? Because I can see it here, but I don't know why it doesn't. Ah, OK. So I have to share it. OK. I can drag it over. Oh, wonderful. OK. So what occurs here? It will start again. So this is chaos. So this is first hope bifurcation. So here you have a limit cycle. This is the duration of the delay. And you increase, you put more and more energy into the system. Here it becomes chaotic. You can hear it, because actually the frequencies are audio frequencies. It starts to become chaotic. And now it's fully chaotic. OK. So you take an Ikeda system. If the typical frequencies are within, if the typical frequency, OK, I will cut this. Starts to drive me nuts. OK. So if the typical frequencies are in the audio range, you can actually, I'll come back to the presentation. No. OK. I will first disconnect it. Yeah, a minute. OK, I'll close this. OK, this is a wonderful operating system called Windows. So OK, close it. Sorry about that. OK, come back here. Wonderful. OK. So there is a second video, but it's very close to the first one. I will not show it. So when you take this Ikeda systems, it's a non-linear system, very high-dimensional. So when you put a lot of energy into it, what you have is you have a sequence of modifications. And eventually, when the gain is strong enough, then you have chaos. So I showed you what occurs when you are in the audio range. So here you have a signal that is below 20 kilohertz. So you can just plug a loudspeaker and you can hear what's going on. But most applications actually occur for frequency ranges that are beyond the audio range. So you can, of course, not hear it, but it's useful for a lot of applications. So the first application we are going to talk about is scale scriptography. So you saw that actually when the gain is very strong, you can have full chaos. So you can use that for scale scriptography. This is what we call the chaos box. It was made by a colleague of mine, actually, a mentor, Laurent Larger. So basically, here you have an emitter. So here you have laser optical fibers. You have some kind of Ikeda system. But very fast, it goes up to like 10 gigahertz. And here you send information. It's encrypted. So this is what you have here. We call this an i diagram. So you have a lot of 1s and 0s. So you have bits here. This is the encrypted part. So if someone comes here, it will only hear the noise. And this is the encrypted signal here. So the receiver has the capability, true synchronization, to retrieve the chaos and listen exactly what is going on. And this actually can be very, very fast. So an experiment has been made in the city where we work in Besançon. So I work here, more or less. This is the city hall. And there is this loop. This is like 22 kilometers long. And we made experiments there. And we could actually encrypt information at 10 gigabits per second. And just to give you another magnitude, it's like 150,000 people talking on the phone at the same time. So it's a huge amount of information. So that's one application of two electronic oscillators we delay. Second example, sorry, that comes to my mind. It's neuromorphic or bio-inspired computing. So everybody here knows how a Turing machine works. This is what we have in our mobile phones, computers, and calculators. But the human brain works very differently. So instead of having transistors or whatever, we have neurons that fire in some kind of like concerted way, have a complex dynamics, and this process information. So more or less, now you have this trend where people try to build what they call analog computers, which function much more like the brain and much less like a Turing machine. So when you take a Turing or Turing Venderman machine, of course, it's great for sequential instructions, very repetitive tasks, high speeds. It's sensitive to input and precision. If you make a new switch one bit, then you just lose generally all the process. And it's sensitive to deterioration. You need to have a very faithful conservation of information if you want the Turing machine to operate accurately. And the example is the matrix product. I mean, I think that nobody here can perform only a 3 by 3 matrix products by head. It's too complicated. For human brain, of course, for a Turing machine, it's very simple. Now, biological brains have all their advantages. Number one, they can work in parallelism. They can learn, they can train, they can improve. They are slow. Turing machines are fast, but biological brains are slow. They have neuronal plasticity, so resistance to input precision and deterioration. And for example, they can do face recognition. So example, if I know you, you can dye your hair a different color. You can shave it. I will still recognize you. But for a Turing machine, it will be a very, very, very difficult problem. But for a biological brain, it's the simplest thing. So here you have one of the most powerful supercomputers. It uses 10 to 20 megawatts. That's the power of one windmill. So it's a lot of power, 20 megawatts. It's 70 years of evolution. Well, actually, the human brain, 20 to 40 watt. 40 watt is the power of a light bulb. It's not much, but several millions of years of evolution. So how do we use optoelectronic oscillators to do computing? So it's a concept called liquid state machine. I won't get too much into the detail. But more or less, there is a theorem that says that if you have a nonlinear system that has a high dimensionality, you can use it to perform recognition tasks. And the way it works is that you have a network. So this is going to be our nonlinear system. You have an input and output. And then you train the system to recognize certain inputs. So we have implemented it under the form of an optoelectronic oscillators. And it has the equivalent of 1,000 neurons. So you can be, OK, human beings, I think, if my memory is good, we have something like 100 billion. So 1,000 neurons is not much. But with these 1,000 neurons, actually, this is the picture taken by the PhD student who worked on this. It can perform the recognition of 1 million digits per second. So yes, it is a little bit dumb. And monotask, it cannot do a lot of things. But what he does, he can do it very, very, very efficiently. And of course, what you use here is the power of photonics. Because photonics is fast. When you use electronics, the problem of electronics is that it's a little bit slow. When you have an optoelectronic system, you use the fact that photonics is fast. And then you use the electronic part for the processing of information. So the other big application, and this is actually eating something like 80% of my time, is microwave generation. So most of you know the history of microwave generation of electromagnetic radiations. Basically, you start from this equation that Maxwell wrote. Actually, he didn't write exactly these equations. He wrote the equations in another form. That is a little bit more complex. But in modern notation, this is how it looks like. And by with a small mathematical manipulation, actually, you end up having the propagation of an electric field and a magnetic field that are perpendicular. But something that is very interesting and that everybody, I assume, knows is that Maxwell arrived to this equation and said, OK, the constant that I have here is very close to the velocity of light. So actually, light is an electromagnetic radiation. But a lot of people doubted that because at the time, the only electromagnetic radiation that they knew was light, visible light. And then there was a hypothesis that actually there were electromagnetic radiations that were not visible, but they had never been detected. So this guy, and Rich Ertz, and of course, it is in his honor that we have the unit of frequency, made this very simple experiment, a very simple experiment. And emitter, receiver, it looks like this. And he demonstrated the existence of microwaves. And when someone asked him, why did you do that? What's the usefulness of it? He's like, you know, it's of no use whatsoever. It's just an experiment that proves that Maxwell was right. So for him, the microwave will have absolutely no purpose. And actually, most of the great discoveries is like that. The people who invented the laser said, you know, the laser says no purpose. It's an invention that is looking for an application. And today, we know all the applications of the lasers. And of course, we know all the applications of microwaves. So just to give you a few, microwave is as a frequency ranging from 300 to 300 gigahertz. Typically, you have, for example, radio station, approximately 100 megs. Microwave oven is approximately 245 gigahertz. Why? Because this is a resonance frequency of water molecule. That's why you can only get stuff that has water. Mobile, phone, internet, Wi-Fi, all these things. Typically, the frequency is going to be between 1 and 3 gigahertz. So actually, we are surrounded by microwaves. But for radars and clocks for aerospace engineering, we actually need waves that are going up to something like 10 gigahertz. So why are we studying this and how the delay can help us with this micro-generation thing? The problem that we have for all clocks, all clocks, since the invention of the clock by Christian Eugens in the 17th century, is what we call the promo phase noise. Basically, when you have a clock, it's an oscillator. So it has an amplitude. And it turns like this. And you can just write it this way. So it has an amplitude, a frequency, and a phase. When you do the analysis, for those that are knowledgeable, you just take the hop-by-fication and you decompose it into the amplitude and phase. You will find that the amplitude is stable. It means that if you perturb like this, actually, the perturbation will be damped out. But the phase will only be naturally stable. So yes, it does not increase, but it does not decrease. If you have noise in the phase, that noise will not be damped. So it means that phase noise is more important than amplitude noise, generally, because amplitude noise will be damped. And that phase noise will increase unboundedly. Not exponentially, but still it will increase. So how does that translate in daily life? So suppose you have a clock, like the clock that is down there or whatever clock that you have. So suppose you have no noise. So at time t equals 0, you have, for example, a needle that is here. After one period, you take this. It makes one round trip. It comes exactly at the same point. That's what you have here. So this is troposcopy, actually. After five periods, same thing after 20 periods, same thing after an integer number of periods, the point will still be there. So here you have a perfect clock. The phase is rigorously constant. Now, suppose that you have phase noise. This is what happens. At t equals 0, you have this. At t equal capital T, here, for example, this clock is a little bit fast. 5t you have, this 20t you have there. And then here, we see that the clock loses accuracy with time. The phase diffuses along the orbit. So this is why our clocks, our wristwatches, our mobile phones, everything, systematically lose time. It's because the phase is initially stable. It's not stable. It's not as stable. It's just in that space that is flat. But since there are always noise, then it increases. Technically, it increases as the square root of time. So just to give you an example, our wristwatches, or the watches, typically a coarse watch, which is what you have in your computer, so everywhere, it typically loses one second a day, which is fine. You can live with that. Plus minus one second in a day, it's OK. A GPS atomic clock has to go to 10 to minus 8. Why? Because when you multiply 10 to minus 8 second by the velocity of light, what do you get? I want to see if you are still alert. 3, OK. This is the precision of GPS with your car. You want a three-meter precision. If the precision of the GPS clock is 100 meter, I'm not sure this will be very useful for you. If you want a three-meter precision, you need 10 to minus 8 because, of course, the GPS computes your position with the time of flight of light. So since the velocity of light is constant, then the clock that is actually deciding about everything needs to be very, very accurate. And of course, if you want to go further, cryogenic clock means that you take the full clock, you put it in some liquid nitrogen or liquid helium. You freeze everything to have the lowest level of noise. And these are the clocks. These clocks take like a room. And with these clocks, you define the universal time. What is the time? Right now, there is someone, some guru somewhere that knows what is the exact time. It's the timekeeper. We need that because we need somehow to agree on what the time is. And then you need the most stable clocks actually to do that. So the way we do it, particularly the research I do, is we use what I call whispering a remote resonator. So I will explain you very quickly what it is. So there is this. If you go to London, you have this dome of the sample cathedral. So people can walk there. So it's here. And since the 19th century and even before, it was known that if you stand in the wall and you talk, someone that is on the opposite side, you talk very, you know, you whisper. Somebody that is on the opposite side will actually understand exactly what you say. So it was kind of like a urban legend. Until actually, some scientists came and said, yeah, you know, it is true. If you come in one side and you whisper, someone on the other side will actually understand. And the first to propose an explanation was this guy Lord Rayleigh. And he said, what happens is that you just have the reflection of the sun in the walls. That's it. It's not that complicated. And actually, you can do that with sound, but you can do that with light. So here, consider here you have like a marble. So it's a pure sphere. Consider it's just a sphere of glass. If you can send some laser light into it, actually by total antenna reflection, your light will be trapped there. And it's exactly the same phenomenon as a whispering a remote. And this is, for example, an example here. So this is a very small sphere. And you see that laser light is trapped there. So why do we use this? And what is the relation between this and microwaves? We can do a clock with it. So if you look at the clock that is over there, actually it has needles like, you know exactly how this works. Now what we want to do is instead of having needles, we want to take photons. And this will give us the time. But of course, this clock will be much faster because, of course, the photons travel at the velocity of light. So this is exactly what we want to do. And for example, this has a lot of application in aerospace engineering. So this is what I do most of the time. And this is an example of this resonator. So this has the size of a coin of 1 cent of euro. It's very small, like 5 millimeters in diameter. It's like a button of a shirt. And here you see the enlargement. And here we sent some green laser light. So the light comes in there. The photons stay there and turn. And after some time, some get out. And the photon lifetime here, so the time if photon stays inside this cavity, is 1 microsecond. OK, to give you a note of magnitude, there is a laser here. And the lifetime, so the time the photons stay in the laser that is inside, is 1 picosecond. So you take 1 second, you divide by 1,000 billionths. This one here, in this cavity, we are able to trap the photons for a duration that's 1 million times longer. And the number of round trips here is not going to be 5, 6, or 7, like in the picture that I showed you, it's going to be 10,000. So this is how it looks so far. Because of course, you have this tiny element. But you need to control that, which is something that is very, very difficult. So we need a lot of apparatus to control it. But we are working very hard, actually, to have it in something that is much more compact. So here you can see that we can put the resonator here, package it, put it in some kind of neutral micro-atmosphere, close it, and here we can control everything electronically. And what we actually want to do is to stabilize it to 1,000 of Kelvin. Why is it important? Because of course, if you have thermal dilatation, if the temperature is not constant, then the disk becomes a little bit wider. If the disk become wider, then the photon takes much more time to make a round trip. And this is an instability in your clock. You want, every time when a photon makes a round trip, you want that time to be the same. But if you have dilatation of your disk, then the time is going to change from round trip to round trip. And you don't want that. So last application that we are targeting with these resonators is that actually these resonators are nonlinear. You can study them with an equation like this. This is called a Lujato-Lefever equation. But for those of you who are familiar, it is an equation from the family of the nonlinear Schrodinger equation. And see here is just the laser field inside. And there you have an instability that is exactly the instability described by Ari two days ago. It's a Turing instability. But instead of having a two-dimensional Turing instability, as the one that he showed here, we have a one-dimensional Turing instability. When you go into the spectrum, this is the spectrum of this instability here. So why do we use it is because actually in telecommunications, the same way that when you wake up in the morning, every radio station has a frequency, 93 megahertz, 94 megahertz, 95 megahertz. In optical telecommunication, every, I will call it station vector, I would have to say channel, every channel has a laser frequency. So here, with just one frequency at the input, you can create a lot of frequency at the output. This saves money. Just with this piece of glass, of nonlinear glass, out of one radiation you can create. Here, for example, you see we have something like seven, six additional. And we made an experiment to see actually if we can use that for ultra-high capacity communications. So here you see we call that a COM, actually frequency COM. Here you can see the COM that we're using for the communications. So we did that with a group in Germany, in KIT. And actually, we could perform ultra-fast communications with it and with the capacity of something like more than 200 gigabit per second. So this corresponds to 3 million fund conversations simultaneous. So consider that in Europe, you have 300 million people. If 1% of people in Europe are on the phone, which is an overestimation, if only 1% of these people are on the phone, all their phone calls will actually be processed by this mode that you see here. So actually, this shows that with these systems, you can actually have ultra-fast and high capacity communications. Of course, if you like music, you can download 10,000 MP3 songs per second. But I think you will need a little bit of space in your living room. And also, it's a little bit expensive. But if you like music, you don't count. So in conclusion, no linear photonics and up-to-electronics is a source of a lot of interesting phenomena. With that, you can explore a fundamental phenomena. Like I talked about the KDA equations, delay, whispering or remote. So you can understand and study very fundamental things. It has a lot of applications for engineering, which is also a very good thing, because it means that it's not too difficult to get funding once you convince people that what you do can be useful to them. It's a topic that is cross-disciplinary. A few months ago, I organized a conference about these care coms with mathematicians. How the people were pure physicists. The other help was pre-mathematicians. And of course, we had a very weird conference where we're sitting. And when the mathematicians start to talk, they're, yes, let's consider a Banach space that is open and closed. They're like, OK. And when we stand up, they're like, yeah, let's consider we have a laser at 1550 nanometers. But at the end of the day, everybody was satisfied with the conference because we could see that what they did as mathematicians were helpful to us because they analyzed the equations that we need. And they were happy to see that actually their very abstract equations in abstract Banach space actually are useful to us, because it enables us to understand what we do. And there are a lot of challenges, both theoretical and experimentally. So what is next? All the time, my conclusion is always the same. From the science point of view, there are still a lot of things that we don't understand. Stochastic, when you add noise, the quantum effects. I didn't talk about the quantum version of these equations, which are very, very fascinating. Technology, of course, when you say this has applications, when you talk to industrials, they always say power consumption, robustness, et cetera, et cetera. So these are points that we have to address. So I'd like to thank all the present and past members because as you can imagine, I did not do that alone. So a lot of PhD students and postdocs have been working with me. I would like to thank them. For these hands on school, these are my two assistants, Jero and Ala. They are doing their PhD in Cameroon in optoelectronic oscillators. And they came here with their experimental setup. So for those who have already been there, they know exactly how it looks like. And I invite the other participants to come when their turn arrives. I'd like to thank, of course, those who pay for all this research. And thank you very much for your kind attention. Thank you.