 Okay, let's start with the next conference, Professor Aristide Marcano from Delaware State University. For me it's a great pleasure to present you, not only because he's an amazing professor, but because he was also my PSD advisor. I hope you also will learn from him as I did in my PSD thesis, and then Professor Marcano, you can start. Thank you. Thank you very much. And thank you very much all the organizers. I see you are a great organizer making this school in optics reality today. And thank you, Dr. Calvo, for inviting me and remembering me because we worked long time in Venezuela and then we immigrated to the U.S., now we are in the U.S. working. But we believe in science, we believe that science is not only for a country, it's for everybody. And this is what these kind of events are about, to say, I am very proud to be here today and honored to be here today, just because 32 years ago I was a student, like you. I was a young scientist in Venezuela. And I met other young scientists and young students from different countries, from Africa, from Asia, from Latin America, even from Europe, even from Americans that were there. That was quite interesting. We have eight weeks. We met Dr. Salan, Abdul Salan was a great person, great personality. We met bright people and they were absolutely believers that science is for everybody. They also were absolutely believers that the humanity, the future of humanity will be bright. We will overcome all these differences that we have in the different countries. Today I work as an American professor in the American university, and a research professor in the Department of Chemistry. But this university is interesting because this is actually an historical black college. So in the U.S. they have a university in the past, they have to assist in the university for black people, university for white people, I think that's terrible. But I really enjoy being there, but today this is illegal, we cannot discriminate. But still we have over 76% of African Americans in our college, and the problems they have is very similar to the problems they can encounter in any country, what we call developing countries. What I remember is 32 years ago the world was very different, absolutely different. And I believe the world from 30 years in the future will be your world, will be also very different. We are expecting a big growth in all, in particular in Africa, because Africa has a lot of resources, young people, good education system, they are developing fast, they will make a point by the end of the century. Asia will be of course dominant because the number of people that live in there, so there will be big changes in the world. So the world is already, you know, there is no one country dominating one another country that is going to disappear, absolutely, there will be only humanity, and that's what we believe, it should be, science for everybody. Okay, let's talk about photo-thermal spectroscopy, so I think Dr. Marine here did a great introduction about some photo-thermal effects, particularly he introduced the Laplace equation for thermal diffusivity process, and people actually, when they do this kind of experiments, they use one frequency, they use only one wavelength, they have one laser, maybe because you have one laser in your lab, you just work with that laser you have, but maybe it's the situation, and you cannot tune, you cannot change the frequency of the laser, so it's very difficult to do spectroscopy, but actually spectroscopy can be made with lamps, and one of the things, in regular spectroscopy today, you do absorption spectroscopy, commercial spectrophotometer, they have just an arc lamp, which is an excellent device, produces a lot of light, and you can just select part of this light and start doing some spectroscopy, and spectroscopy is actually a big achievement of humanity, because thanks to spectroscopy we developed quantum mechanics, because people will discover that atoms, they don't take any kind of light, they take some particular bands of light, and why that's happening, so we change the way we think about matter, and we change everything. Okay, you have any question about our university? This university is in the state of Delaware, state of Delaware is in a small state, it's one of the 50 states of the US, and is that small, it's difficult to find in the map, but you just look for Washington, it's nearby, on the Atlantic coast you will see the Delaware state, nearby Maryland, so everybody knows Maryland because of Baltimore and Washington, but nobody knows Delaware, and Delaware where we work is a nice city, it's a typical small town, American town, it's very nice, very safe, so I've been actually fundamentally happy there, I'm working, and we developed this method, so we are going to talk about that today, and mostly there are two things, two ideas here, two devices we can actually create, these devices are created by light itself, so it's absolutely optical method, we use detectors just to detect light, we do not detect temperature, nothing like that, we can do that with the so-called optotherm, it means that you have a temperature detector you detect, but here we are going to develop some devices where the light detects itself, it's absolutely all optical device, one important thing about this method is there are two things, one is the thermal lens, and the other is the thermal mirror, I think Dr. Marin has introduced a little bit about the thermal mirror, is that deformation that you have there, but some people don't look at that as a mirror, they just look at that as some changes in the refraction index, actually it's a mirror, it's just that you are doing some changes in the reflectivity of your surface, and that produces some interesting signals, some interesting effects, and the thermal lens, one particular thing about this method is the sensitivity, so the sensitivity of this method, just for the first years of development of photothermal effects, people understood that you can measure very small amounts of contaminants in water, you can measure with this method the absorption of water indivisible, because when we see water, we see the water is transparent, we don't see water as an absorbing anything, we believe it's just transparent, but we know that water, you have a lot of water you will see scattering, and you see that has bloom, so the bloom is just the scattering of light you have, so you don't believe there's an absorption there, but with this method we can measure, I'm going to show you that, we actually have the record of measuring absorption of water indivisible using this technique, and then we're going to see, use this, why this we need that for, one important thing I have learned in America is that what you do should matter for people, so you don't do that for yourself, you do things to make some products, you do things to satisfy some needs of the society, and I think it's a good approach, so that's the point, as the Americans they have good engineering programs, excellent engineering programs, and people are developing devices, it's very dynamic, very competitive, it's very difficult to get funding, but it's very interesting at the same time, and we use that for characterization of materials, so right now there is some kind of new kind of materials, you know, every day people are discovering, you know, graphene was the latest, it's Nobel Prize material, before that was the C60, the fluted in was other material that people invent, something that has particular properties, and graphene will be still, it's still being studied carefully, there is a lot of interesting optical properties of graphene to discover, and graphene is the material that launch a new kind of approach, now you can work with two-dimensional crystals that have the thickness of one atom, and this is absolutely new material, and that is making a revolution, and for that there will be a lot of applications in doing capacitors, where I believe in the future that capacitors will be that big, that simple, that you just go to Walmart, go to the store, you buy your electricity for a month, that's what I believe is going to happen. I remember when, because I was advisor of Umberto that was a long time ago, but also a long time ago, I don't want to say I know, I still feel very young, but a long time ago I was given a lecture about LEDs technology, and I told people we're going to have LED TV televisions, at that time we didn't have that in the 80s, and we're going to have LED screens, very big screens, everybody started laughing, didn't believe that it's going to happen, and that's funny that today you have this technology right there, you just buy $200 and you get a nice LED TV, and there are these technologies on your cell phone, everybody has that, and saying that this is going to happen was difficult to believe at that time, but you need to see that in order to, you need to just to get the education in order to see and prevent that that is going to happen in the future, so that's one of the reasons I really believe the future is bright, is I believe. One thing is, one important thing is okay, photothermal spectroscopy has been considered some kind of absorption spectroscopy, but absorption spectroscopy is something that's very well known, you just measure transmission, you see this, you measure how much light you have before the sample, you measure how much light you have after the sample, you take the, you call that transmission, that depends on the wavelength, you take the logarithm of that, the chemists they take the investment logarithm, the physicists they take the natural logarithm, it's the same, but they define differently and there's a big confusion, but it has a lot of applications, and they start, we have these applications anytime we do a blood test, we do a blood job or work, blood work, so they actually, what they do is absorption spectroscopy, and they compare it for different wavelengths, how much of this vitamins we have, how much of these, all their atoms we have, whatever, and minerals we have in our blood, okay, but for the thermal spectroscopy, and I want you to convince about that, it's something different, of course it can be worked as an absorption technique, has been used as an absorption technique, but it's giving you something additional, so I want you to convince that after these lectures that absorption, for the thermal spectroscopy, it's a spectroscopy to be developed, because we do not have commercial devices that do photothermal spectroscopy, you just pluck a bottom like a spectrophotometer and you get your spectra, you don't have that, I do that with my own hands, the spectrophotometer is handmade, is homemade, and it's difficult to work, because we do not have enough technology and resources to make it that automatic, but it can be done, and then I will convince you that this is absolutely new kind of spectroscopy, and can be used for characterization of materials, in particular photonic materials, materials that have application for solar applications, they can be used for analysis of plot, they can be used for analysis of tissue, human tissue, I believe you can do imaging for this, people are doing that, people are doing microscopy, for the thermal microscopy, and they are getting a fantastic resolution, people have reported that you can actually measure one molecule with photothermal spectroscopy, not one nanoparticle, one molecule, they have reported that you can measure that, basically about five years ago. What is photothermal effects? What the thermal effects is, that's the general definition, so anything that happens when you have interaction with light, so you have interaction with light, you have the matter, you have electromagnetic wave, they interact, they exchange energy, there are some changes in matter, there are some changes in light, so because they are exchanging energy, and anything that follows, after you have the absorption of that, there is a generation of heat, and one interesting thing is heat, it is generated always, even if you have a highly scattering sample, even if you have highly fluorescent sample, you still have a lot of heat there that can modify the properties of matter. One of them is, for example, you can modify the, you have the expansion, this is the effect of the thermal expansion that you have, you can have, when you have some heat, you can change the volume of your sample, and there is a coefficient called volume expansion coefficient, and defining that way alpha t sometimes, and you may have some changes in the refraction index, so you may have also changes, directly changes in the refraction index, you may have changes in the absorption coefficient, you may have any kind of changes because of thermal effects, these are all a generation of the cold photothermal effects in general. What are the two basic characteristics of these photothermal effects? First is the universality, universality means that, as I say, anytime you do any kind of experiment with light, you will have some heat being generated, you will have that, and sensitivity, so you can achieve unprecedented sensitivity with this method, and at one time I have some discussion with some physicists back in my country, Venezuela, and they actually like meteorology, they did interferometry, they believe, the people who do interferometry, they believe interferometry is the best, because that was invented by Michelson, the interferometer of Michelson, and that was a very sensitive technique that allowed to measure the speed of, changes of the speed of light, but they discovered there is no change, and there was a big revolution, as you know, in physics because of that, and the resolution, the device was so sensitive that there was no way that there is a change in the velocity of light, it was very sensitive, and I have a discussion that says that the interferometry, you cannot surpass interferometry, so there is something to believe among physicists that interferometry is the best, you cannot surpass, and they say, listen, this photothermal method is a phase method, interferometry is also a phase method, I'm going to talk about that, and the phase method, they have the same sensitivity, and they didn't believe it, and we did an experiment, we compared the two methods, and we actually showed that it's the same, you have similar sensitivity to interferometry, but the simplicity of the experiment plays in favor of the photothermal method, and we will see about that, but in any interaction, as you see, it took that for the internet, you have the laser ablation, is what we have, we have a big laser, we have people do what they call laser lips, laser-induced breakdown spectroscopy, just make a plasma there, too much with one pulse, you have a lot of energy, very short time, you make multiple ionizations of your method, and then these ions, they recombine, and then they start making a million light, you can do some spectroscopy of that, but the point in any kind of, in particular, in this kind of interaction is evident to generate a lot of heat, and this heat has been irradiated everywhere, into the sample, into the atmosphere everywhere. It's a famous Jablonsky scheme, Russian scientist who actually invented that, Jablonsky, and I believe he was from Polish, but actually this guy was Russian, and actually this is a scheme, it's a quantum mechanical explanation of model of complex organic molecules with light, it's something that interests us, organic molecule like chlorophyll, hemoglobin, or any dye, they actually say in these molecules they have some kind of electron, which is called the pi electron, which is an electron more or less free, and this electron moves in some kind of box, so this is a classical quantum mechanical problem of the electron inside the box, and the good thing about that is you can solve that and show the basic principle of quantum mechanics solving this problem, it can't be solved exactly with the extraordinary equation, but he showed that this electron actually can has a spin recombine, the spin of this electron can recombine with the spin of the rest of the molecule, get the zero spin and they get the singlet states, these are the singlet states, what, these are the singlet states, there is no laser, you get, okay, I have no laser anyway, what do you see there, S0, S1, S2, those are the singlet states of the electron, it means these states with spin zero, those are actually electronic states, S0, S1, S2, and there's a lot of different states, those are vibrational states because the molecular has some vibrations, oh great, thank you, maybe it's more powerful, good, thank you, so you have this S0, so these are the singlet state of the electron in that big molecule, and they can be actually excited, these electrons to another level here, S1, or even higher, we don't write more states because if you put that, so the molecule breaks apart, so people say well then they put S4, S5, because it's too much, so you break apart the molecule, and then they have other states called triplet states, triplet states, they are with the spin one, so there is a general rule that there's a forbidden rule that you cannot make transition from triplet to singlet state, so this triplet state is usually metastable, it means it stays in their energy, they are accumulated for a long time, it's just a nice system, people discover die lasers at that time, the point is that in this you can say okay you have the absorption of light, and then you have what they call relaxation, which is this wave, waving arrow that we have here, so it looks like a relaxation, where is the energy that's going, so the energy is going into heat, so we have just heat, so we have collision with other molecules that they just start moving faster, and they get an average temperature, which is bigger, and you have some heating, so here you have heat being generated, here you have some heat being generated, and then you have some relaxation from these singlet states into triplet states, also you have some wave, there it means that they call that inter-sistant crossing, there's a difference of energy, where the energy is going, into heat, into heat, you may have also transitions directly from S1 without doing anything that's heat, you can generate fluorescence, so they call, or you can generate heat as well, so in all these transitions, any time there's a transition, there is certain amount of energy generated has heat, it means this is nothing else but the second principle of thermodynamics, because we see heat is just energy that you know is decomposed, we say it's decomposed, it's degraded, we say that it's the second principle of the thermodynamic of the system. All right, I did this small calculation, so you can do it, it's a good exercise, so imagine that you have, today you have this device called the nanodrop, but you have this device can measure micro, microliter, a drop of a microliter, so imagine that you have one microliter of water, and you have one atom that is absorbing this light, so it's absorbing this photon, and you see the point is that this atom, these all these processes of relaxation inside the molecule, they are very fast, they happen in this in picoseconds, phantoseconds, sometimes even faster, but it could be even, maybe the the longest time, maybe it's about tens of nanoseconds, something like that, but this is very fast, and the point is that if you have that light being irradiated, so the electron, the atom will take the energy from one photon, relax, give the energy to the surrounding molecules, then he's ready to take another photon, he's ready to take another photon and he can do it that multiple times, so far you have the photons there, so you have continuous illumination of that, and then you will start heating your sample, and I make just a simple calculation, you have that, the time it takes the atoms to release the energy to the surrounding water molecules, it's about 10 to minus 10 to minus 15 seconds, it's very small time, and you have, then you have what happens with the heat, stays there, but there is some diffusivity, so you have some diffusivity, diffusivity is the normal reaction of matter just to get the equilibrium of temperature in the system, because of the Fourier law, we have that trying to get an equilibrium, eliminate the gradient of temperature, and this diffusivity takes too long, it takes too long to happen, it could be between milliseconds, even seconds, sometimes I do experiments of the diffusivity, it takes over 10 seconds, it's a long time, so imagine how many photons, these only atoms can absorb, so you can absorb 10 to 8, 10 to 13 photons, one atom, and when I calculate how much this is going to be in terms of increasing of temperature of that microliter, it's about 10 to minus 3, which actually could measure with the current technology of temperature, we're talking about one atom, one atom, you know, you can imagine how many atoms we have in one microliter, I will see the, do the math, you see this immense number of atoms of molecules of water you have in that microliter of water, there is another fact about sensitivity, so the good thing about thermal effects, they're accumulative, they accumulate over time, so you are sending the energy all the time, you are heating your sample, of course if you wait long enough, the sample will start boiling, you can heat it, you can do the same thing, you're doing the cooking, you're just heating that, but now you're heating with an expensive heating with a laser, you don't need to do that anyway, but anyway, you have other fact that sometimes for the thermal effects, they are of phase related to the changes of the refraction index, and you can actually write the phase here, so the phase of the wave will be just 2 pi, the length of propagation divided over the lambda, the lambda is the wavelength of the light, and then it multiplies by this factor which is very small, but it's still okay, why, because L over lambda could be very big, could be 1 million, because lambda you have 500 nanometers and L could be 1 centimeter, and then you get actually a big number too as well, so these two factors make this technique very sensitive, so from the first years of discovery of the thermal effects, you know actually thermal effects were discovered by accident, if you know maybe you hear about that, I don't know if somebody say that, it's about the people who were working in the 60s, they were trying to make short pulses and they used dyes to make a passive monoloking, make passive monoloking in the laser, and these dyes they observed something very strange, they observed that there is a photothermal effects inside the dye that destroys everything, so they hate that thermal effects for a long time, try to eliminate that, here for example is some applications that this is taken from the book of Baikovsky, he's a professor for the U.S. university, and he has a good collaboration with different Latin American scientists, and he's an expert, actually he's an expert in photo deflection technology that was actually discussed here, and he just said that this is, you can do interferometry, here for example you have an interferometer or you could be Fabry Perot interferometry, you can just imagine that you have your light going passing a lot of times, and then of course anytime it passes you have the thermal effects, and you can accumulate them, so you can have an L very very large, some people have developed this multipass system, you can have one kilometer, one kilometer over the wavelength is immense, an immense number, people use, you do that for doing absorption measurement of water, I know in Texas there is a group doing that, but now for the thermal you will increase a lot, I think Dr. Caprera has been using this similar idea to improve the sensitivity of the technique, I don't know if you have that here, the students will have that opportunity to see this experiment in place, you have the thermal lens, so they work, what they happen is that you have a light, and this light you know the light is not everywhere, it's just a beam of light, it's a Gaussian beam, so in the center it's more intense, in the borders in the winds, the winds it's less intense, so you have some kind of distribution of intensity, and that produces a distribution of temperature, is distribution of temperature some kind of thermal photograph of the laser, and that of course changes the refraction index, that produces a lens, in liquids this lens is usually negative, so you have why, because in liquids you reduce the density, you have, you reduce the refraction index, you're reducing the density, that makes what they call a local the focusing lens, and then you have some spreading of that, sometimes it's very impressive, you have an argon laser, happen to have a big laser, you put in any absorbent liquid, you will see a big what they call the thermal blooming, it's like the light is making an explosion of that, you have also multiple fringes, there's a beautiful experiment, you have something like that, people have studied that for a long time, and then you have this other photo deflection, it's spectacular, it was explained here for Professor Marine, other you have diffraction, you can create diffraction gratings in the system with two lights, you make thermal gratings, and these thermal gratings actually produce diffraction of light, it can be used that, it's some kind of order effect, it's a typical non-linear effect, you see in this excellent book by Belkovsky, you can read, this book was published almost 20 years ago, 20 years ago, it's still a very actual book, like this book, this is thermal lens, it looks funny, but this represents a lens, please believe me, it's not a lens, it's just you are focusing a beam of light, suppose a Gaussian beam, and you know that in the center you have the maximum intensity, and of course here you have in the center of the beam, you have maximum intensity, but then to the whims, the intensity goes to zero, and of course it will produce more changes on the density, more changes of the refraction index, with reduction of the refraction index, and then you have there far away from the center, you will have nothing, and then you create your, the focusing lens in your system, that's the thermal lens, what happens is the light will fill that change, and because it's a phase change over the propagation, and when you do the diffraction problem far away from the center, you will see changes in the, so this phase changes in one point becomes very high amplitude changes at the long, at the far field, what they call the far field, that's a typical optics problem, we're gonna talk about that, and this is the mirror, people do that, a mirror is, the good thing is some materials are not transparent, so you cannot pass, you cannot pass the light through them like metals, you cannot pass light because absorption is too big, and you know the metals they can, they do not allow the electrical magnetic field to go inside, but they get a little bit, they get a little bit there within one wavelength, so you need to have a four layer one wavelength to measure in transmission, that's very difficult, and but usually you can do some interesting experiment, what they call you just send your pump laser, and then you produce some deformation of the surface, surface will be distorted, and this distortion will produce some phase difference between the light being reflected from here, and the light being reflected from here, and then you can collect then this phase difference, you need to do the integration, this mathematical speaking is not an easy problem, but a lot of people have thought about that, actually this problem has been solved basically with zone approximations, and you can actually discover a signal over that, in my method, in the method we developed actually over the years, we use two beams, we use a probe beam because it's a lot more convenient, and we're going to talk about that, so this is the probe laser that tests that pump beam, and we can actually are monitoring what is happening to this probe light, this is the thermal mirror effects, we're going to talk about that, now let's talk first about, because we're talking about principles, let's talk first about thermal lens, the photo thermal lens, so it's actually you can, it has a first approximation, that's the first approximation, you can consider that what do you do in, is because you create a phase in this plate, this is a phase plate, so you have initially this field, you just stamp it, that plate over that, because you are filtering, and it's a phase filter, you just add it, that function, this is the phase right here, which is the pen of the temperature, and then this beam has been modified, because the phase is different now, this phase is because we say this 2 pi lambda L changes of the refraction index, and this changes of the refraction index can be due to temperature, so we have that property, this number is about 10 to minus 4 centimeters, 10 to minus 4 Kelvin degrees, degrees Kelvin, minus 1, this in the gradient, it's a very small number for all samples, but it's still big enough to do the experiment, and we will see that, we are going to people to develop the theory for that, if you want to see the people have been working for over 40 years on this thermal lens effect, and they have now they are making a lot of applications, and you know what is the difference, the difference is because in the past, in the 70s, when I was student, all lasers were homemade, you need to build your own laser, or the lasers that were commercial, they cost half a million dollars, so you need to be a rich scientist to get the laser, or usually you build your own laser at that time, today they are commercial lasers, but it has been a revolution in lasers, so now you can buy an excellent laser to do this experiment for 300 dollars, so you are interested in doing this experiments in your home, in any conditions, that's probably one of the cheapest, no linear optical experiments you can do, because you can buy a small laser, it has about maybe 100 milliwatts is good enough, and I know, I can give you data about some Chinese company that produce these lasers, and I have very good relation with them, and they are very competitive, because if you go to the big guys, they want to charge you 10,000, there is no point, you are crazy, it's too much money for that, if you want to do an application, you need to make it affordable, you need to think about people, so it's not about just only science, you need to make a real product, that's the good approach that I learned in America, you need to make it real access to the problem, and yes you need to make it accessible, so low price, low price is very important, that's the Gaussian beam, it's a problem, you know the Gaussian beam, everybody believe Gaussian beams are what is happening, actually it's just an approximation, it's this second approximation, after considering that beam is just a line, you see the geometrical optics, after geometrical optics, the second approximation is to consider that this is a Gaussian beam, but this is still an approximation, so actually the real beams are quasi-Gaussians, you have a good laser, so they have this M factor, M2 factor, you have an M2 factor close to one, this is this very good laser, but usually some lasers they have M factor of 10, that's about laser, M2 in two factors, it's something that you study optics, people can explain to you the opinion of this M2 factor, but this is the Gaussian beam, this is the properties of the Gaussian beam, they're basically two properties, it's just the waste in the center, because the smallest amount you can focus that, and it's because of diffraction, you cannot focus more than that, it's related to the same concept in quantum mechanics, that's the principle of uncertainty, that you can actually measure position and velocity at the same time with high, with top precision, you cannot do that, here happens the same thing, you cannot focus the beam completely, and there's a limit, this is A0, and there's a second parameter called the Rayleigh parameter, it means this distance where the diameter becomes about 40% larger, square root of 2, larger, and that is the Rayleigh parameter, and of course you can this AZ with the radius of the beam changes with the position, so you want to do this kind of experiment, you are having your sample, and you are moving your sample here, it's not the same intensity that moving the sample here, because here you have this more focus, here it's less focus, so you have the same amount of power until in a smaller or bigger area, and you will have more intensity in the center, and of course the thermal effects will be very big here, thermal effects should be smaller here because you have more gradients, you will see that thermal effects depend on the gradient of the temperature, this is the question, it's impressive when you see that the first time, I don't know you have started that Gaussian beams, but there is a theory, very well developed, I think there are a very nice book of optics by actually an Italian author Pietrotti living in America, an Italian living in America, an excellent book called Introduction to Optics, has a good explanation of this, has something for a typical general course on optics, and it has, this is the amplitude of the electrical field of the Gaussian beam, it has an amplitude in the center, has the radius here that changes with position, this is the radius that changes with position, this is the equation for that, and this is the, here you have this radius that produce this Gaussian behavior of the intensity, and this is a phase that we have, this is a constant phase, and this is a phase that depend on the position, and this phase depend on the position, the transversal coordinate there that we have, and there is other parameter called the radius of curvature, this radius of curvature, we are at the waist, the radius of curvature is infinite, because this is like having a sphere with the center in the infinite, and then it changes sides, you have the radius of curvature, when you have z will be the position of your, the position of the horizontal position you have, this z will be this, z here, this is 0 here, this negative z, this positive z, this is more or less the theory of Gaussian beam, something I recommend you, you want to do optics, you want to check that theory carefully, try to understand everything, the Gaussian beam, and make a plot, just if you have any program, make a plot of that and have some fun, having different Gaussian beams and see how they behave and why, because some people have trouble understanding the rally parameter, they don't understand that, now it's very clear, when you do the experiment, you do the calculation, it's very easy to learn about that, here you have you take the square of that, you take the square of this, of course all the phase terms they disappear, so they don't matter for intensity, and you have, this is the intensity of the beam, we have normalized that over the total power, this is the power p0, the power you measure, when you put the beam on the detector, you measure the power, it gives you milliwatts, watts, whatever you have, so there are different kind of detectors, they are calibrated detectors, and they are not that expensive, so you can buy actually, and forget about this detail, how to calibrate the detector, you don't have a good detector, but anyway, as you can see this function has what they call an axial symmetry, axial symmetry, there's no dependence on the rotation of the axial angle, but of course this is an approximation, you want to do some theory, you will see that actually the beams are a little bit more complex, there is some dependence on and people have making interesting discoveries about that, but in our theory we are not going to consider these details, because they will modify, we are going to consider that we have this axial symmetry, but again I want you to see that that has an approximation of reality, and of course we can solve the Laplace equation, the Laplace equation is that the question that Dr. Marine was talking about, this is the Laplace equation, here we have the differential equation, partial differential equation, which is here you have the derivative in time, here you have the spatial derivative, this is the nabla square which is the Laplacian, it changes the double derivative in space, and it has the diffusivity coefficient that I think Dr. Marine explained a little bit about this equation, but this is a basic equation was actually discovered 200 years ago, developed it, and people have been solving since when, since then, and just lately people have started solving this equation by hand, but today they are excellent programs, people they have, I don't know you have heard about console program, I don't want, I don't work for console, don't think about that, I just, it's a good program, it solves this equation in a variety of situations, so they have the solution for this equation in a variety of situations, but solving this equation requires knowledge of mathematics, so you need to know how to solve the differential, partial differential equation, how to solve that, there's some basic principles, if you just have a course on basic methods of mathematical physics, you should cover this one of the most important equations, besides the equation of the wave, this is one of the most important equations you should learn how to solve, here you have this part, which is the part created by the absorption of light, this is the absorption right here, you are absorbing that light, and now we are considering that all this light is being transformed into heat, or heating the sample, this is our source of heating, and this is the power, we just discovered the Gaussian power, and here you have the capacitance, the heat capacitance of the sample and the density of the sample, that's the typical equation, and people have tried to solve this equation, I think this is the cylindrical coordinates, I don't know if you don't, just for you to remind, but it becomes a little bit complex problem in mathematics, but we are not going to discuss that here, but I think you can see if you want to teach that, it takes at least one lecture to explain how to solve this, but this is the Laplacian, the Laplacian or the Napa square in cylindrical coordinates with axial symmetry, so you can actually plug this into that equation before, and try to solve that using partial differential method, and here is the general solution of this kind of equation for the difference of temperature they have here, it's a function of position, transverse of position, it's a function of time, here we have eliminated the, we are considering that in z, in propagation along the sample, we have the same thing, so we have, we call that the thin lens approximation, so there is no changes in the profile of the beam inside the sample, so inside the sample, the beam still remains with the same radius, more or less, it's also an approximation, and then you have this, the so-called, everybody hates them, the green functions, the green functions, so this is the green function they say, this is a modified Bessel function, so you need to have that culture of mathematical physics to know what is the modified, modified Bessel function, so you can actually google a little bit what is the modified, you will see at the plot, but today's computers you can just put that function i0x, you get the function, you get the function, it's beautiful, it's like a sinus and cosine, we don't remember all the numbers of sinus and cosine, the same thing, so today's computer they provide this function immediately, without any trouble, I have this coefficient right here that provides how the evolution of heat in the sample is happening, and here you have, here it's just giving you this trick, but when people are using the papers they don't say about that, they're giving you this trick, they use this equation which is the table integral to solve that, this is a table integral, it has to solve one integration, so we have two integrations here by zero to infinite in the coordinate, space coordinate, we have this integration, then we have an integration over time, the tau here, and we get this equation is just a trick, a mathematical trick, there are some tables of integrals, so you can actually find, you cannot find, you just google tables of integrals, and then it pops up, we didn't have that technology 30 years ago, and that was impossible to have that, anyway, and we can just forget, just a function, imagine this is like a sinus, and don't think too much about the complexity of the mathematics behind, and then you have, you can obtain the solution, and it doesn't look, it doesn't look also very nice, that's the typical elliptical function, it's also a very well studied function, there are, people have also, make tables of these functions, but you can calculate with a simple program, and use that matcat, you know, it's a very simple program, matcat, matlab, you can use any kind of mathematical problem to solve this equation, and it's easy to calculate, it's very well behaved integral, so it's not a crazy integral, and you have, this is the solution I found, I calculate myself, changes in temperature, when you have water, you're using 30 milliwatts of green light, this is the jack laser, double harmonic of the jack laser, you have water, and you have 30 milliwatts, and you create the thermal length, you create the temperature, the changes in the temperature, here start at the beginning, this is normalized over the size of the beam, it's because the size of the beam is normalized, if you have at the beginning, very short time, you're talking, you're looking, this is 100 milliseconds, it's not that short, it's a lot of time, 100 milliseconds, and you see, it still very resembles, resembles the beam, the Gaussian beam, so it's like a photograph of the beam, and then you have the spreading of that, it's growing, and it's growing because there is absorption, so you are heating, you are increasing the temperature there, but then you have the diffusivity, it's spreading this signal, it will continue to grow until boiling, but the shape doesn't change, so there is some time, there is some time when the change becomes stable, it doesn't change, it just grows in amplitude, and this seems that the, because the signal we are going to detect is proportional to the gradient, so the signal gets stationary value, so it doesn't change any longer after, after some while, but just after a while, we're talking about seconds, you know, it means that it's the typical thermal effects, I just use the thermal diffusivity for water, and the lasers we have, lasers, you can see 30 milliwatts is something really, it's not big, the refraction index, also you can actually, people have, how it changes the refraction, actually people say about that, they forget that this is also an approximation, you need to do that, do the Taylor expansion of this, we're going, but this is because this is 10 to minus 4, 10 to minus 4, this is 10 to minus 8, so you just forget about that, just eliminate, but water has an important term there, it's interesting that water behave a little bit strange, because water is an important matter, an important sample, and it has some contribution for this term, right there, it makes the calculation extremely difficult, you cannot do that, because we want to make it simple, so we just skip only this term, and that's it, and forget, it's just again an approximation we're doing, which is also important to remember, because maybe some other effects can happen, in particular, you are working with water, and that's the value for ethanol, I think, the value for ethanol is very well known, that's multiplied, I don't know what the plus, it's multiplied, here okay, here you have your thermal lens, you are creating that, you create that delta F, this delta 5, delta 5 is proportional to the temperature, we just calculated, and we can actually measure the change of the phase, in general you can say okay we have, we can have changes because of the refraction index, or because of the length, because of the changes of the volume, we can have changes because of that, and these two factors in particular, you're working with solids, they are important, if you are working with liquids, this part is not that important, and if you are working with the mirror scheme, the first part is not important, so it changes, the contribution of these two parts is different, dependent of the sample, undependent of the experiments you are doing, but suppose we have continuous excitation, we have a laser, which is always there, and let's try to solve this problem, so we have okay, so we define the difference of phase, then you have, difference of phase will be the, the value of the refraction index at some point are, in the transversal coordinate, z is the position of my sample, so this is where my sample is, and time is the time, because thermal diffusivity, there is dependence on time, and this is the value of the refraction index at the center, and because there are some changes, you have a gradient there, so you will have these changes in the phase, and that changes in the phase, you can actually see taking the solution of the, of the temperature, you can see that you can write this equation this way, and of course, resembles, resembles the, the thermal lens of the calvary, here, when you use two beams, you have two beams, we introduce one interesting concept here called the mode matching coefficient, this is just the relation between the radius of the probe beam, and the radius of the pump beam, so we say that the two beams, they may be very similar, in this case we say that we have a mode match, but in thermal lens, it doesn't make sense to do that, because now you see that the thermal lens spreads very much, a lot, so you can actually take advantage of that volume, taking a probe beam larger, and make it a more intense signal, getting a better signal, and that's, you get mz, which is the factor, this is the square of the radius of the probe divided over the square of the radius of the pump, and you call that the mode matching coefficient, in all our experiments, this coefficient is over one thousand, four thousand, it's very big, in other experiments, when you do single beam, you cannot do that, you have matching coefficient of one, so it means single beam experiment doesn't give you the best sensitivity, it's not optimal, but still people do a lot of single beam experiments, which is okay, because the method is still sensitive, and you can calculate this phase here, which is the amplitude of that phase, it's proportional to the intense, to the power, you're being to the absorption coefficient, to this parameter of the sample, and the thermal conductivity that they have here, so you have actually, you know, if you know all these parameters, you can measure all thermal conductivity, or you can measure the absorption, but in this case, this is not absorption, this is the absorption that was used for heating, but you measure absorption in the transmission experiment, you can have some discovery, and that's absolutely different, you may have a sample that has very big discovery, this is the single beam experiment, the good thing about that is very simple to do, the bad thing about that is not that sensitive, and it's not that easy to understand, since people can get very nice experiments, and then you can do just hand scan, your sample around that, so you do the z scan, but that gets some picture of that, this picture is something like a z of zero or something like that, but people say, why in the center, why in the center you have a smaller signal when you have more intensity, you have more intensity there, it's because the fraction compensates the effect, you need to go from the positive signal to a negative signal, you need to go through zero, so whereas you have a maximal intensity, you do not have the best signal, so that doesn't mean this single beam experiment, this is designed, it's not the best design, but imagine that you do an experiment where now you use two beams, you can modify, you can use lenses right here, you can modify the rally parameter of the probe and the rally parameter of the pump at your will, and you can make it in your sample, you can make the probe being a lot bigger than the pump, and see, here you use a filter to cancel the pump, and then you are observing distortions of the beam of the probe, right here you just put an aperture, and then behind the aperture you put a detector, because you have changes in the focusing lens, the beam will be doing something like that, if you are chopping it increases and it spreads away, and then you get changes of power going through this detector, and that's your signal, that's the positive signal you can actually measure, but advantages of having two beams, as we can see, first you get highest sensitivity for this fact, now you can take advantage of the beam where it's the most intense, because you have more power, more intensity, time dependence, you can do time dependence easily, interesting is that you can actually pulse, you can use a pulse laser nanosecond, and then after some microseconds you see the thermal lens growing, it's impressive, you see, you see the pulse, you have this time dependent experiment, and then you see the signal growing after the pulse is already gone, because the heat is there, it's just evolving, and then you see this evolution with the probe, it's a beautiful experiment, spectroscopy, and that's what they interest me, you can do spectroscopy, you have a tunable laser, you have a tunable laser, you can actually do some tuning and start doing some photothermal spectroscopy, and that's an interesting thing, you can actually do UV spectroscopy using a visible detector, because you are detecting the probe beam, not the pump, so if the pump, if the pump beam is UV, it doesn't matter, so you are detecting still the probe in the visible, so you do not need to have UV detectors, which is a big deal, and they're very expensive, infrared detectors are very expensive, some sort of them, very important, you need to cool them down sometimes, because you have a lot of thermal effects there, and you can actually do infrared spectroscopy without using infrared detectors, that's an interesting approach how you do that, I'm thinking about submitting the proposal for that, but some people don't believe in that, so it's not easy to convince people, and you can do different, you know, now you have versatility, you can start inventing, I want to do this one this way or the other way, I change a little bit my laser, whatever, I can change the color of my lasers, whatever, I can do a lot of different experiments, and we have invented this method, it's called the optimized two beam experiments or mode mismatch, what we do, we just collimate the probe, we collimate the probe using a telescope, it's a collimator or a telescope, it's funny, just a couple of lenses, but you buy the telescope, they charge you $1,000, I don't understand, a couple of lenses, just two lenses, and just think about that, before you spend $1,000, you think buy a couple of lenses, that's good, that will work, that will work, and you have your probe being collimated here, passing through the filter, and then you have the pump, the pump is being focused in the center right here, and here you have maximum intensity right here, and then you will see maximal signal, now you don't get the z behavior there, you can get a peak, which is the result, it's the same, but at the same time we discover when m is very big, this is the best way of doing the experiment, this is the best sensitivity you can get, and this is mathematically correct, I can show you why later, but we published that, I guess you can see Cabrera there, it was the part of his PhD I think, at that time, that was a good paper, and here we define our signal, okay, we define our signal is the light of probe light passing, when you have the thermal lens minus the light, when you do not have the thermal lens normalized, and the good thing about this definition is that it's a unit less, there is no unit, and this signal we can actually show is proportional to the absorption, so you measure this signal, you are measuring the thermal absorption, what do we do, we do, you don't have a break, how many minutes I have, are you tired, wow, you don't make a break, okay, okay it's good, for the sake of simplicity we will consider that, ah, that's a good approximation, it's actually invented by Brazilians, physicists, they, what they do, instead of doing, this is the power you measure in the detector, so you need to integrate the intensity of the light over the detector surface, but you can have very, very small detector, you need to have a very small detector, it's like you don't need to calculate anything, just put r equals zero, and that's it, you put r equals zero, you just calculate it here, you put r equals zero in your calculation of the probe amplitude, and it works perfectly, and it's very easy to calculate that way, and it gives you a very similar result, if you are, you want to do very careful things, of course you need to make that integration, and sometimes you can do it, you have enough computer power, and it's not, it's no big deal, good, how the point is, how to calculate this amplitude at the position of the detector, so remember we are doing this experiment right here, so we have our, this is your sample right here, at position z, and then it goes into the detector, of course this is exaggerated, it's very close, actually take more than a meter to place the detector far away, and then I see what is happening in the, in the center of that, just taking r equals zero, just taking the center, and what they do is just, I need to solve the diffraction problem, and this is a typical diffraction problem, when you have, you know the, what is your field in the position of the sample, this is the field in the position of the, you know this is the plane of the sample, you know that there you have to modify amplitude with the face, you have a face there, and then you need to calculate what is going to be the field at some distance d, and this distance is big, compared to the size of the beam, and this is the so-called Fresnel approximation, you do the Fresnel approximation, it's a typical diffraction problem in the course of optics, but it's not an easy problem to solve, it has some mathematical complexities there, but I'm not going to do, deal with that, you can find in any book, but anyway, just need to believe me that this is, this is the result, has been test, but actually you can calculate that's the amplitude there, and you get this result, you see the pain of this factor here, we have this constant three, this constant, the pain only on properties of the probing, so the pain on the rally of the position of the probing, the pain of the rally parameter of the probing, and the pain of the position of the detector, which is d, that d right here, you can actually see that v is just a number for the experiments when you have m very large, this v is about 20, 40, it's a number, and it's multiplied by the unit, complex unit, and then you have your face, you just calculated there, this is your face, I'm good, now you can plug this into the, into the new computer, you can do that, and the computer start giving you the amplitudes, how this happened with the amplitude, you know it's a complex function, you can have the face, you can have the amplitude of that, but the detector can detect only the square of that, so the detector detects the square, and then after a lot of simplifications that you can find the tails of this, this is the best vapor, this is one of the first papers that I actually provide, I think I, when I obtained this formula it was a little bit different, but I like it the way he published, this way I think it was before me, and I'm just making this citation right here, of Shane and Snoke, some people from England, this Chinese professor in England, and this is the paper, you can, it's a very good paper, I recommend you to have it if you like thermal lensing, it's a paper to have, and you can actually get a lot of simplifications and calculate for small absorptions, for small phases, you have the small, I mean, smaller than 0.1, 0.1, smaller than that, 0.2 is still okay, 0.3 maybe it's too big, still for small samples you can actually find, you can find this behave, you have the amplitude of the signal right here, you have the absorption, the power of the, of your pump laser, and these photothermal parameters, and then you have these arcotangents, and you know that the arcotangents has a maximal value, the maximal value of the arcotangents is pi, you can have more than that, it's limited, so it has, it's only pi, and that happens when you have mode mismatch experiment, that's the reason this is optimal, so you optimize your experiment making this factor bigger than anything by doing this experiment with collimated beam, and what does is surprising, nobody does the experiment that way, besides Cabrera, now Marina, I convince him to do it that way, some Brazilians, nobody else, nobody, why? I don't know, but this is the best way, it's just math, you cannot make arcotangents bigger than pi, I just found the conditions where this arcotangents is equal pi, and this more or less default mode mismatch situation, where you have M very big, that's simple, seems to be a lot of math, but at the end it's not that difficult, and here you have two, besides the amplitude, here you have this parameter which is the mode mismatch, now we know it's a big number, and then we have these parameters, time evolution, it's called the time built up, the time built up experiment, you want details of these calculations, we did in the past, there was about 10 years ago, we published that, and in that paper Joseph B, this is the, when you do mode match, I mean the typical z-scan experiment, when you do z-scan experiment, you see this typical z, and you see the evolution over the time, you see, you are making, this is at the beginning, and then you have more hit, you have more signal, more hit, more signal, and the shape doesn't change at some distance, so that you have actually what you see at the center, where you have maximum intensity, you have small signal, zero, because of diffraction compensation, so it's not the best way of doing the experiment, but if you do the mode mismatch, you can get this way, so you get small, then it's growing, it's like more natural, it's something that you understand better, you see, why you have that z, why is zero when you have more intensity, that confuses, that confuses a lot of people, why you have zero when you have the maximum intensity, where you should have the maximum signal, you have maximum intensity, and here you have, you have that, when you do mode mismatch, you have the maximum intensity, where you have maximum signal, and this is calculations made, we're using math cat, very simple program, and that formula I just showed before, and this is a real experiment, real experiment, we just have our pumping here, being focused, we measure point by point, that's very difficult to do, there's a lot of work doing that, we measure point by point, we measure the radius of the beam, they use the razor method invented by Boyd, this is very difficult, today they use a device, but at that time we didn't have that device to measure the, you can use a CCD camera to see the radius, you can do that today, but at that time we didn't have that, so we measure point by point here, and then we have probing, the probing was collimated, this is the collimation that you have here, coefficient m over thousands, so they have that, and this is a real experiment for water, what's interesting is this water experiment, you see this is the probe, what is happening with the probe beam, your probe beam is, it has a continuous light, it's continuous there, when you inject the pump, it starts the thermal lens being built enough, it builds up, it goes to zero, it goes to some value, and it gets the stationarity over five seconds, so you get the stationarity here in water, what they have there, and what is interesting is the water that was done with the red light, 632, the probe was 632 and the 532 was the green light from the laser, and we have two millimeters of water, are giving you a big signal, but it doesn't look impressive, it's not that impressive, but let's do the following, let's do this, I can just from this calculation for this measurement, I can take my signal and say that this is T0, this is T, I take this minus T divided by T0 and I get my signal right there, so it seems to be a little bit noisy, it doesn't look so impressive, but you can do the following, you can use a filter to eliminate that DC, you don't need that DC, you only need the DC value for calibration purposes, and then you can take that AC component, amplify, if you amplify the whole thing, the amplifier's gas becomes saturated because it's too much current, but if you amplify just a little bit, you can amplify a lot, and you get a very good sensitivity, and here you have what they call for water, this signal is not purey, this is an experimental signal, it looks theoretical signal, and you see this is the chopping of my pump, I have been chopping and the signal is negative, it grows up because I'm taking this AC, amplify and then average, and I do that, and you can get that sensitivity, this unit is the noise, the signal to noise, they have signal to noise 5,000, you see, and here you have, and here you have just, okay, arbitrary units, whatever, but the noise is very, very small, and then I notice when the cell was without water, I notice it doesn't have a signal, and the signal was from the cell, from the glass of the cell, but the signal was very big, I was able to measure this signal is about 1000 times smaller, I was still able to measure with some noise, right there, and the signal has another sign, has the coincidence with the sign of the pump, it's positive, so in glass, the changes are positive, but in water they are negative, and of course when they are, you have in glass and water, this small change doesn't affect the water results, we're talking about 2 millimeters of water, that's impressive, I show this result to Dr. Boyd, but Dr. Boyd is a professor from Rochester University, very well-famous scientist in nonlinear optics, he told me this is too good to be true, he didn't believe it, okay, publish, at least we published that, this publish is okay, we published in a blackface letter, here you have for water it's negative, you get your z-scan, you see it perfectly, and this is the absolute value of the signal, it's very small for water, but we are able to see almost a perfect signal, and this is the signal from glass, and it took us a while to do that, the averaging, to do that, I average over thousands of pulses, and then I get that 10 to minus 7, it means that I can measure a signal, 10 to minus 7, less than a million times smaller, a million change, these are, you can change the frequency chopping of water, start playing with that, it changes with the time, you can do different kind of experiments, so I think I can go ahead, here you have, what I wanted to show you here is that the signal to noise ratio for water is in the thousands, in the thousands, and we are doing that with a small laser, so you cannot do that with normal spectroscopy, if you put water, one cuvette of 1 centimeter spectroscopic cuvette in a normal commercial spectrophotometer, you cannot measure anything but in the infrared, only in the infrared you can get something, but you get sure it's below the sensitivity of that device, because it's 10 to minus 4, it's very small, and you can measure there maybe 10 to minus 3, so it's still 10 times smaller than the sensitivity of the regular spectrophotometer, so you cannot do, water for you is nothing, it doesn't have absorption, it has, very small, but it has absorption, and say what do you need that for, here there's another property I want to show you, it's interesting property, is that if you have scattering, you have scattering, you cannot produce heat, if the light is scattered, so the light is not trapped by the matter, so it means that the scattering light is not affecting the effect, it means that for the thermal method it's scattering free, and people didn't believe in that, just do an experiment, let's say let's do an experiment, and here we do an experiment, we measure the turbidity, turbidity means that you have water and you see it's not clean, and you have some particles there inside, it's turbid, and they measure the turbidity, it's just similar to absorption, you just take the, how much light is passing through, you take the logarithm of that and you call that turbidity, now it's not absorption, it's just losses because you have something in water, some particles that scattered light, and here you have turbidity, zero, here to have turbidity 8.6, I have this my signal, how I increase turbidity, I have some, I think I use metal here, water, I use some sample, water or some maybe I can put some dyes just to increase the sense, the signal was big here, the signal was big, I put some something additional to increase the absorption, and I have, I start using some micro spheres, latest micro spheres that increase the scattering, if you put too many of them, that becomes like milk, when you see milk, which is white, that's actually scattering, it's not the real color of milk, but you have the scattering of light, white light, scattering, it's mostly scattering, yeah, we are, our skin is in a scattering sample, and that's actually designed by nature, so we don't need to be hit so much, we're gonna die, we have too much temperature, and we have, we don't know how to, we have some absorption, but most of the light, visible light is being scattered with our skin, so what we have is just scattering of light, that's interesting because okay, we're now, we have a technique that is scattering free, okay, now I increase the scattering 15 centimeters, this is similar to the small, the skin, the skin has a scattering property, it was 20, 20, 30, it's similar, I still get the signal, noisy, because it's reduced, but their signal is still there, I can actually, okay, it means that maybe I can measure my signal through my body, and start doing some imaging of that, you know, my bones, of course we need a little bit better technology, but anyway, I did these experiments with just micro spheres, and show that there's no problem with the scattering, here you have measured, here we measure the signal as a function of turbidity, you have turbidity of over 10, you start making that signal, what is happening here, you have multiple scattering, multiple scattering, and that erases a little bit the gradient, because now you have bigger, the gradient becomes smaller, that at the end the turbidity can affect of course the thermal signal, but you still have a signal, but if you do transmittance, transmittance right here, you see a signal, which is four order of magnitude smaller, four order of magnitude smaller, it means that you cannot see anything through transmittance, but maybe in photo thermal you still can see something, that's an interesting idea, can we do imaging of bones using visible light, so we don't need to use x-rays, something for new generations to think about, maybe we can do it. Here you have in highly turbid samples, I guess take milk as a turbid sample, and it has turbidity of 10 right here, so you see the thermal lens, thermal lens signal going down a little bit, and then suddenly it's been destroyed, but it reduces about 100 times, it's still there, smaller, but it's still there, so you can still detect in milk, which is similar to normal skin of people, and here this is the real signal, you have this signal, the detector we still have, you can see we have even passing through a good turbidity we still have, we published that also, this is the reference, you can check this paper 2013, just to finalize this 15 minutes, I would like to introduce this photo thermal mirror effect, and talk a little bit about that, and what we have, as I say, that we have a palm bean, and it creates a distortion, but this distortion is very small, it's nanometric, it's about one nanometer, maybe less than that, it's very small, but it's good enough, it's good enough to produce changes in the face, remember, not because of the lamb, that's very small, we still can have a good face effect because of that, and then you can start to do a similar experiment, but now in a mirror configuration, what do you need that for, when suppose most of the samples, minerals, you have, if you have transparent sample, no big deal, so you can do it in transmittance, but if you have metals, you cannot measure the absorption of metals, it's very difficult, because metals absorb everything, and minerals, mostly the old materials that are not transparent to light, but you still can analyze the surface of the material, you still can have surface and make some distortions of that, and get the signal, you do the same thing, you solve the same Laplace equation, similar to that, but now you need to solve this, you need to add another equation, it's called the equation of the thermo-elastic distortion, and it's a beautiful equation for people who likes math, but here you have double the gradient of the gradient of divergence, and then you have the nabla here, wow, and then people, when they do the reference to this equation, they cite a book, which is a book of a Polish author, it's so complex, it's so difficult to understand, I don't want to reduce merits, but I found a book of Landau, you know, Landau was a famous Russian physicist, Nobel Prize because of his contribution to superconductivity in the 60s, and Landau has an excellent collection of physics, what in the book of elasticity, about the theory of elasticity of Landau, I found this equation, it has a beautiful demonstration of that equation, absolutely elegant, beautiful thing, it is a difficult equation, but this equation happily with computers can be solved, but we didn't count with the group of Brazilians who solved this equation, found some solutions of the equations, and they actually published a lot of papers about that, and these papers are very difficult to read, you need to know math very well, you need to know that, if you like math these are the papers to read, if you want to understand better this, but they use boundary conditions, one of the boundary conditions is that far away from the laser there's no changes in temperature, and the other boundary conditions is there's no stress, this is the stress tensor, is that pulling this distortion, so there's no stress, the stress is just where you have the beam, that's what is the meaning of these difficult concepts and sometimes people get, because it's a tensor, it's a tensor concept, so it becomes a little bit more complex there, but anyway, the idea is that it's a free space, it's a free space, and when these boundary conditions, you define the phase difference has the, you have something like that, you have your beam is coming to the center and being reflected, or it's coming from the wings, it's being reflected, but there's a difference between them, you can actually find this difference distortion, the difference in the distortion when you have r equals zero at the beginning, and you have what r equal r far away, and you have some difference in phase, that's similar to the thermal lens effect, but that is a mirror right now, and this is the scan of the experiment, so you have your palm beam, and then you create that distortion, then you can probe, you can have a probe being that test, can test that and modifies doing a very similar experiment, but now in the geometry of reflection, and it may have a lot of applications, and these are the references for this Brazilian group, I just throw them, you are interested in how to solve this equation, and it's a beautiful thing, I think these papers are something about a method of mathematical physics, it's a good advance of this, it's a group of physicists in Brazil, but they collaborate with other people, and Baikovsky maybe is somewhere in all these, oh here you have Baikovsky collaborate with him, he has that in one of these papers, they're really good papers, but when you are doing an experiment, you read these papers, they don't understand that, it's too difficult, it's true, you need to do have something practical, and what they did, okay, I spent my time, I modified a little bit, it results, and I found this solution, and this is the phase you generate, they found the equation, the way they found it, I'm not going to talk about that, but this is the phase you generate, you get that, here there's a function f of eta, now you see this equation is beautiful because there is no unit, everything is unit less, there is no unit, so you don't worry about the coefficients and anything, you have a time dependence tau, which is you have this time built up, you have this dependence g, which is the distance divided over the size of the probe beam, the radius of the pump beam, you have this, and you have this integration over eta, right here, and any computer can take it, and I did that, you can actually calculate that with a simple map, with matcat again, you can calculate that easily, you want to buy this program, I recommend, for students it costs about $100, for commercial applications it costs thousands, but we cannot pay thousands, we pay $100, it's okay, I say I am a student, they pay $100, they don't care, and we used that solution, we just published a couple of years ago, and we're still working, and we just finished one, another work, interesting, I couldn't talk about that, a little bit, and here you have the phase amplitude, it's proportional to this 5, which is not this alpha g, this is not the absorption, this is the thermal elasticity, thermal elastic coefficient, you can measure that, this is the Poisson ratio, so the Poisson ratio when you have changes in the volume, so the solids, they expand in this direction differently, from this direction, they expand differently, so this relation is called the Poisson, that depends on the structure of the crystallographic cell, the pain of that, the pain of the atomic properties, the way it's expanding this direction is not the same as expanding this direction, this is called the Poisson ratio, this constant is interesting, but sometimes it's difficult to understand when you see these equations there, they put this Poisson ratio, you should know that, okay, anyway, I just put in a way you can actually understand, you measure the amplitude, you get something proportional to the thermal quantum gel, what is the thermal quantum gel, this is the number of photons used for heating divided by the number of photons absorbed, you have 100 photons absorbed, maybe 80 photons were used for heating, and that's it, you get the energy of it, that's quantum gel, and that's what you measure in this method, that's interesting thing, now you are not measuring absorption, you are measuring the thermal quantum gel, actually, just actually it's a new parameter, nobody knows nothing about that, it's absolutely a new parameter for characterization of samples, but you have this time without, you can build, you can measure the thermal diffusivity, but now, in a thermal mirror, it's good because you can measure at some distance, you don't need to go far, you have some materials far away, you have a nuclear reactor, you want to measure the thermal diffusivity there, you are heating too much the reactor, you can do it, it's remote, you can do it that remotely, that's interesting, and you can have, again, the amplitude to fill, again, we use the diffraction theory right here, we use the same definition for the signal, the signal when we have the phase and we do not have the mirror, so we just chop, we cut the bin, put the bin, cut the bin, we just chop some of the little bits, and we do the experiment a lot of times, and that's what I get, that's the theory, that's what Matcat gives me, the theory, I get this theory for different, this theta, it's just this phi zero, it's the amplitude, and I take the same amplitude, but I take different rally parameters for the probe, and I see if you get bigger rally parameter, you get bigger signal, but the time behavior is not affected, the time behavior is not affected in the way you are actually focusing your probe, that's a good thing. Here you can actually estimate with the theory, the signal has a function of this phase, if the phase is too big, it becomes a little bit nonlinear, is you are getting the phase of one, radian, we're talking about the radian, and if you are getting the phase of 0.1, still here, it's still linear, so you can actually work, you just need to check, when you do your experiment, you want to have some signal that is proportional to the quantum gel, you need to make sure the linearity of your signal, you just make sure the signal is not that big, and that's the photothermal mirror spectrophotometer, this is actually a new device, and what we have here, I don't know if you have time, it's over, five minutes, okay let's finish that, okay we have the pump here, we just go to a reference, then we take that pump, we focus on the beam, and after focusing, I just put a block, I put a block here, so this beam is not needed longer, you can put another mirror, you get more effect, but you don't need it because the lasers have enough power, and then you have the probe laser, you can use the laser as it is, or you can use a collimator if you want, but the laser is already collimated, it has about one millimeter, a couple of millimeters in radius, it could be considered a collimated beam for this experiment, and then you have your beam here, it's being reflected back, then take that reflection, and take, I just increase a little bit with this lens, this, when you put this lens, it's actually, it's like having a small aperture, because you just amplify the amplitude, so the other is just the amplified, and you amplify that, and you get your, you see, detect your signal, here you got, and here you have some results of your experiment, and what you have, this is the result for a glass filter, you have this black glass filter, it has a lot of absorption, and you get your reflection, you get a very big signal, this signal has been normalized to the, to the stationary value, the stationary value is somewhere here, so it grows very fast, and then goes into the stationary value, and the red light is the theory, the model, so you can actually, but in here you have the time in units of TC, so you actually can have, this is unit less, you can measure your TC, just fit your experimental data until you get a good coincidence with the theory, but this red light actually is a universal curve, it happens for any sample that follows those equations, you find something that does not behave like graphene, I don't know, maybe it's different, maybe you have some quantum effects, but if you have a classical material that corresponds to the solution of these two equations, the thermo-lastic equation, the Laplace equation of the diffusivity, they should behave that way, if they behave differently, there is something new, so you have a paper, and here you have nickel, nickel, I just have a couple, several metals there, nickel you got the same thing, but it's different now, you have different TC, but here's a unit of TC, and this is normalized over the stationary value, and you get again the same behavior, it's the same curve, so all times the red curve is the same, but this is nickel, this is glassy carbon, glassy carbon is, it's a carbon which is polished, and it's used because it has a good, it's used in electrochemistry because it has, forward has a note, because it has good electrical properties, conductivity, and things like that, it's a nice device, it's a nice material actually, and I like it, it has a good signal, and you get, I did for a lot of different glasses and things like that, and then I measure here what you call the values of this built up, and this is what the red lights did theory, and these are my experiments, so the experiments for different copper, platinum, it didn't behave so well, but it's still there, nickel, titanium, quartz, glass, and then I have a collaborator from Stony Brook, he gave me this material, this prosion tetanate, this material is used to control nuclear reactors, and it's a very interesting material because it has properties, a very high absorption of neutrals, so you have a nuclear reactor, it's going out to control it, to put the, that, try to get back into control the reactor, and you see it's, you see one, two, three orders of magnitude, a different, you go from glass to copper, I would like to have gold, gold somewhere here, but they don't have gold, and you can actually make a calibration curve, and then from that curve I can calculate how much is the diffusivity of the, this prosion tetanate, I have calibrated my experiment with known values of materials, and of course it's a lot of work, you need to do carefully, there is no robot, the robots are my students and myself, there's no robot there, so work a lot to get these, these points, you know, and we measure also the amplitude, the stationary value, so the stationary value, it should be proportional to the phase divided over, over the power, so remember the phase is right here, right here you have divided over the power, you get something that is proportional to the quantum gel multiplied by these thermolastic parameters, right here, you have these thermolastic parameters, so you divide this over this, and you get this last picture they have right here, so you have the line, and you have the, the, for different materials from copper up to glass, and then I put this prosion tetanate, and this prosion di-tetanate, and it happens to be, they have differences, they have good, good thermal diffusivity, they are similar, but the thermolasticity on the properties of elastic properties, they are different, and that's interesting because it's more flexible, it's the other, and that could be understood why this is happening, is because this prosion di-tetanate, it has more atoms, it's more compact, it's more difficult to, it's more difficult to expand, and you can actually see that it broke, because you have more titanium, you have two titanus there, some 5-7 oxygens in this area, it's more difficult to deform that, that, that's the reason you have here this, almost, almost 5-6 times, this is a good number, so that was this, and we can make a conclusion, and the conclusion is that good, photo-thermal lens and photo-thermal mirror are versatile techniques, are very sensitive, and you can use for determination of absorption and photo-thermal properties of material, we can make that general conclusion, and the use of pump probe configuration allows the implementation of the spectroscopy, is what we're going to talk tomorrow, we're going to talk about the spectroscopy experiments we have done with these two configurations, and why I should show tomorrow, why this spectroscopy is a new spectroscopy, it's not regular absorption spectroscopy, I hope I convinced you tomorrow about that, thank you. Thanks Professor Marcano for this complete and nice lectures, and now you have time for some