 And I would like to introduce Professor Maria Luisa Calvo with a talk, a lecture on laser speckled informatory theory and application. Thank you very much. Good morning. So when we started design the contents of this college, that is dedicated to advanced techniques for bioimaging. One of the main idea was just to try to have some kind of a convergence between what we were seeing here in the room in terms of fundamental basic knowledge and maybe some theoretical approaches with what was needed later on just to be seen in the laboratories. For that, you cannot choose very complicated techniques for the simple reason that we should be needing a much larger infrastructure, which for the moment we have not. And then you see for what you were seeing in the precedent days in the laboratory, that essentially this is dedicated to techniques that are assembled with not a very heavy infrastructure. So one of the techniques that is very convenient for this type of orientation of the course, the contents, is just the laser speckled phenomenon. And this is what we are going to talk now about these fundamentals. So we're going to introduce some discussion about what it is, the concept of laser speckled. How can we interpret it in terms of mathematical formalism? And we'll be discussing some classical techniques, in particular the difference between what we named stationary and dynamic laser speckled imaging. So first, do not forget just to focus the idea that this is an imaging technique. So what you're just going, in any case, to analyze is the image of this speckled phenomenon. We will mention about this image forming system and mention briefly some applications in biomedicine. Because now there is a special term for this. This is biospeckled. It turns out that this, as I'm going to mention now, that speckled phenomenon is very interesting for biomedical applications. And one of the reasons you immediately are going to understand is that it is kind of assembling a non-invasive technique. So when we are just dealing with techniques in, for example, diagnosis techniques or some following some patient treatment, it's very important just to look for non-invasive. And speckled could be in one of these under certain conditions, of course, as I'll be mentioning now. So those are the contents of this slides. Introduction, the speckled formalism. I'll be just talking about the first order statistics, which is the easiest way to understand speckled as a random interference phenomenon. I'll take the special case of an image forming system. And then I'll be giving some details about what it is, this dynamic speckled in image forming systems. I'll mention also some equivalent phenomenon that can be observed in nonlinear optics. And as I say, I'll give some examples of biospeckled. And in the end, what I'm going to show you is the explanation of what you are going to see in the experimental laboratory. So those of you who are going to have the laboratory this afternoon, groups four and five, you're going to see this experiment in speckled. And then the other groups will distribute it in the next coming days. So what we understand by speckled, this phenomenon was observed, starting being observed early in the 60s of the 20th century. And the main reason was the introduction of these new highly coherent laser sources, which is the laser, one of the many collections of laser sources that we know today. So having these new laser sources with such high coherent, spatiotemporal, coherent properties, what happened is that they were revealing like a new phenomenon that was not observed before, but maybe from in precedent centuries by Lorelae and some others observing phenomena in nature, in atmospheric phenomena, for example. So there was this pioneering work done by Joe Goodman and Chris Tainty, in which they introduced what it is, the fundamentals and the basics of speckled. And then we can say that this speckled is observed when we are illuminating a surface with a highly coherent source, as it is the case of laser. This is what you are having here. This is the surface. And then as you are seeing in the profile of the surface, this is kind of a random irregularities. This is due to the rough. And then if we illuminate with a highly coherent beam, what we are going to have here is a phenomenon of multiple scattering. What is the meaning of multiple scattering? What we are meaning by multiple scattering is that each one of the individual scatters are scattering light. But since we are dealing with very highly coherent beam, then we are going to observe also interference between these scattered fields. Therefore, do not pay any attention that in a speckled phenomenon, we are having a very complex explanation because we are having multiple scattering. Of course, here in light diffraction, we can also interpret. And we are also going to have interference. And the structure that we are seeing when we illuminate a rough surface with a, for example, with a helium neon laser. Well, here is in blue, but imagine that it's in red as you are going to see later on. This is the structure of what we are seeing. If I illuminate now with my pointer laser, this white page, this white page, it's having a certain rough. This rough is what it is for, but this is scattered centers. And then you'll see, I mean, maybe from far you are not going to be able to see it. But if you approach, you are seeing not only what it is saying, the projection of the beam in this screen. But you are seeing also a kind of a halo surrounded in which what you are observing is the speckled structure. So how can we demonstrate that speckled phenomenon is arising from the coherent properties of the laser source? For that, what you have here is what it is known as the concediness experiment. The concediness experiment consists of having a double slit, John double slit. And then after the plane of the double slit, you can introduce a diffuser. And then, of course, you'll have the observation plane. You have here the double slit plane to illuminate. This has to be a laser, of course, highly coherent source. Then you introduce here a diffuser. And then some point after the double slit, the interferometer plane, you'll have here the image of what you are going to see. If we observe here what we are having in this plane, of course, this is a speckled pattern. It's a speckled structure, which is looking a little bit more into it. It's having this random structure. But notice I am not sure if you can just see it from there very well. So you can notice that inside the speckled structure, inside the areas in which you are having this bright spot here, you are seeing fringes. Because of course, what you are having is this double effect of having the interferometer experiment and the diffuser that is producing the speckled. So you are having this structure. This is a speckled. But in these areas inside the brighter spots, we are seeing fringes then telling us that in this phenomenon, we are dealing with a certain degree of coherence. If now I'm moving slowly the diffuser, now the diffuser in what I'm explaining you here, the diffuser is static. Now if I'm start moving the diffuser, it turns out that what happened is that the fringes and this structure disappear. And what we are having is just this incoherent beam illuminating the image plane. The fact that we were moving the diffuser was what we know as temporal decorrelation. So what we were doing was to decorrelate the temporal coherence properties of the beam. And therefore, we are not able to see the speckled anymore. So we fix ideas that from an incoherent source up to a very highly coherent source, I can have quite a degree of coherence in sources in a very highly coherent source. My speckled will have good quality. But of course, I could be using some partially coherent source that will be less quality. And with incoherent beam in what I'm just explaining to you now, I am not going to have speckled because, of course, I'm going to have a scattering with this beam. But I am not going to have constructive and destructive interferences. So you are seeing here a more detailed speckled pattern. This is interaction of helium neon laser in a plastic slide. So you are seeing here like a complex structure. Here at the center, this is what the equivalent to that. It's kind of a diffraction pattern as you're seeing here. But then going to the other areas of the pattern, we can observe that there is forming brighter spots. Here is when the brighter spots appear when we are having highly constructive interferences and darkest spots, for example, here, it's when we are having destructive interferences. So in all the speckled structure, what we are going to have is a distribution of the irradiance in this plane. We are going to have a distribution of the irradiance. And this distribution is a continuum of values that are going from the minimum up to the maximum the brightest spot. So because of this appearance, let's say this is a chaotic chamber, it's introduced in the name of speckled. We notice too that I was showing you now here with this red emission. Our eye is able to observe the speckled. So a speckled phenomenon is becoming visible to naked eye. But I would like to mention that these days there is a lot of work in speckled with other type of sources, having, of course, a certain degree of coherence as it is the case for the x-ray. And this is having a very interesting application in biomedicine too. How could we interpret this chaotic phenomenon of speckled? Let me mention that in the beginning when this phenomenon was observed, it was observed when someone was dealing with image formation with a highly coherent source. And then it wasn't good because we do not want such an amount of information in the image plane. And the speckled was introducing kind of a noise. This is making a misinterpretation of what was really what is in the image plane. Therefore, in the beginning, it was like some undesirable effect for what it was this image forming system. And they were looking about eliminating the speckled in one of the ways to do it is as I was mentioning to you before as by introducing a temporal correlation with the source. But later on, it came to with all these contributions of works, one start appreciating that a speckled contains a very complex information about the structure of what, of the rough surface. And then it's starting a very important area that is introduced in methodology. And later on, also seeing that it can be applied in certain areas of biomedicine. So it was passing from, this is a good example of how science can turn for some phenomenon that we can think that this is not good. I don't want to have this effect. It's just turning out that now we are using this effect for other purposes. So how can we interpret? We need a formalism to interpret what is happening. For that, I'm going to follow the Goodman treatment. We have this very well-known random walk model. Random walk model is very well known, not only in the area of optics, but it can be used to explain chaotic processes in some other areas of what we want to interpret. And then what we are going to assume is that we have this single scattering of the coherent light. And what we are going to have is a collection of particles. Remember, each point of the rough surface acts as a scatter. We are considering just assuming what kind of scattering are we having. So we assume that the scattered dimension is much greater than the wavelength of the illuminating radiation. And then we're also considering that we are having here is the scalar field. We are assuming that there is no depolarization in this formalism. This is very important, because we know that when there is a reflection, we have light, and there is reflection. I'm sure that many of you have observed that there is a slight depolarization. So in this case, we are not considering such an effect. And we are going to work just with the scalar field. And of course, it's not in the formula, but you all understand that this is going to depend on the spatial coordinates and time, and going to what it is in the right side of the equation, same for the amplitude, and same for the phase. So this is the starting point, and this is the starting point. And of course, you see the phases moving in between these values, and it's statistically independent from the amplitude. You are seeing here that what we are having is a discrete superposition. What we are telling is that each one of these scattered phenomena are being superimposed in the form of this summation. And then it is the number of the independent contributions. Then what we are going to study, what we are going to say, is that the skater in amplitude has associated a probability density function. Apart from what we are seeing here for what I was explaining, in this random walk model, we can represent in the complex plane. And then what we are having here is the fuzzer. In this case, in red, this is the constructive addition. And in this case, this is the destructive addition. So in between these two, that can be very variable walking random processes as represented in the complex plane. So how can we define this probability density function? For that, I'm going back in time. Maybe it's going to be soon 100 years from the contribution of Lorelae in 1990 in the Philosophical Magazine. And I'll try to read. It's not well written there. When the phases are at random, the resultant amplitude is indeterminate. And all that can be said relates to the probability of various amplitudes, to the probability of various amplitudes or more strictly to the probability that the amplitude lies within this limit. So this is a probabilistic interpretation. We cannot assure what is going to be the value of the amplitude. Therefore, we need to introduce some statistical description. And in his work, Lorelae was defining this probability density function, as I say, assuming that the phase is independent from the amplitude. And what you are having here is the radial profile of the joint probability density function. So this is a case that we consider that it's having a radial symmetry for convenience. And if you observe, this is the phase component. And if you observe, this is the zero order Bessel function. And n is a finite number. It's a finite number. And what we are having here in between brackets, this is the average over the ensemble of all the scattering amplitudes. If we observe in detail this formula, we notice that applying the Fourier analysis, we know that this is a Bessel Fourier or Hankel transform. Then what we are having here to interpret this probability density function is a Bessel Fourier transform of what it is this ensemble of amplitudes in this finite number of events. So what happened is that this formula is having difficulties to be solved analytically. Because I insist that here n is a finite number. So we cannot consider that it goes to infinity, just to make, for example, an approximation. We need to maintain that n is finite. So this is very complicated to be solved from the analytical point of view. And also even if you would like just to, of course, you can try numerical, that's always possible. But it is also introducing difficulties. So we are going to some simplification of the interpretation. And in what it is, this is statistical properties of the amplitude. We know that we can consider the process of being Gaussian or known Gaussian. So of course, we are going to choose the process to be a Gaussian one. And this is what it is the first order statistics. And this is very convenient. And for what we are just needing after that, just to arrive to characterize the speckle, it's convenient to use this Gaussian statistics. And we are having here the first order, second order moments. And it turns out that when we now introduce the assuming that this number n tends to infinity, we are seeing here in the second equation, this goes to 0, it goes to 0. So this is just having a constant value. And then we can interpret the process having a Gaussian statistics. And what you are having represented here, this is from work of Chris Dainty. This is those are real measurements. It's a histogram taken from a speckle pattern. It's 23,000 events. And this is what we are obtaining for the histogram. It turns out that it behaves as a negative exponential. So this is what we are going to consider and what all that is coming after that. The ones that we can settle the formalism and the type of statistics that we are going to apply, then we can go too far just to see what the type of applications that we can do. And then notice that here when this probability density function is 1, that it corresponds to the case of the intensity being 0. So let us take an example. I was mentioning before that in an image farming system we initially were not interested in speckle. But later on analyzing the speckle given us interesting information about the structure. So what we are having here, this is the simplest case of an image farming system. You are having here the object plane, just one lens and the image plane. And then in the object plane, what you are having is the rough surface. So what you want to check is what is going to be the structure in the image plane. We remember that we are working under the conditions that the scattered dimension is larger than the wavelength. And in this particular case, since we are having just one lens, this is going to coincide with the aperture pupil, the pupil plane and the entrance pupil of the system. So what is going to be, how are we going to calculate the maximum size of the image speckle? Say that we are going to have here quite a distribution as you are having here in these examples. This is for the particular case in a hot Sherman sensor. So we are going to have here a kind of certain complicated structure. And we want to know what is going to be the size of this speckle image in the case of the lens is aberration free. For that, we use the theory of diffraction. This is what we are using because we need to take into account the resolution of the system. And if we remember the relay criterion, and then we need just to know what is happening with the response of the system to punctual sources. But it is equivalent to say that I want to study the point spread function of each scatter. So each scatter, according to the theory of diffraction of light, each scatter in this system is having a corresponding point speck function. This is going to be in this plane. And in here, we are going to have this sort of contributions. So we are taking the airy disk as the diffraction pattern reference. And we define the size of the speckle. You see that is proportional to the wavelength. What we are having here is the magnification of the system. And what we are having here, this ratio, this is nothing else but the f number of the system. So under normal conditions, if we are using this formula, we can know what is going to be the size of the speckle in the image plane. There are restrictions according to what we know about the spatial resolution. Their restrictions is that this formula, as far as I was reading from some contributions as this one, it does not hold for high resolution optical systems. Because we can have been considering that in high resolution optical systems imaging that we are having one micron of the size of the speckle. And then we are going to have problems to interpret here in terms of resolution. So we can characterize this through the theory of the diffraction. And another important parameter is, as in any other process with light and image formation, is the contrast factor, the speckle contrast. The speckle contrast is the ratio between the standard deviation and the intensity average. So that when this speckle contrast is just one, what means the deviation in that case is equal to the average. And then it's saying that the speckle is fully developed when the contrast equals one. And if not, this value goes from 0 up to 1. We are going to talk about the speckle contrast later on in some of the cases that we are going to consider. So this is just to finish with this part of the, oh, sorry. I forgot to tell you something I think important that I was missing. You don't need this system to observe speckle. I was taking this case because it's easier for you to understand the way we are just introducing this definition. But of course, if you remove the lens, imagine that you remove the lens and you just have the object plane with this rough surface and then a certain detection plane after that, you can also observe, of course, speckle. This is the far field case of speckle. If I introduce the lens, then I know that this is just going to be acting as the geometrical of this tell me as an image permit system. But if I withdraw the lens, I always have the phenomenon as well. So I was saying that to finish with this, what can we interpret to the light of what I was telling you about the point of speckle function? So we are having in the image plane is a collection of point of speckle functions that are going to be randomly distributed and in which you are seeing here what will be kind of the projection and in which, of course, you have all sort of possibilities that this is resolved or not resolved or constructive or destructive interference. OK, so sorry, since I needed to use a PDF file and you are not going to see this movie, I'm very sorry. So what we were explaining before is just what we named static speckle. Speckle means that this diffusive surface is static, it's not moving. But this is what will be here and here that you cannot see it even in this light. It's all very pitiable, I'm very sorry. So here what you will see is that this is all moving as what I'm just doing here is all moving and they are saying that this is a boiling structure. So when we are having the case of this surface, this scattering medium is changing with time, it's evolving with time, then the speckle pattern also evolves, of course. And then what we're obtaining is what it is named, this time variable speckle or shortly dynamic speckle. So we were just seeing that for static speckle, we are just having this definition for the contrast and in the case of the moving speckle, we need to redefine the contrast. Why? Because there is fluctuation of the intensity. We are having this boiling structure here. So we are going to have changes, the intensity fluctuates, and we need to extract information about this speckle contrast, what it is going to be the level of blurring. So it's going to be a blurring in the structure because of this moving. That can help us to set up very interesting applications in biomedicine. So in the case of this moving speckle pattern, the definition is the same, but notice that now we are introducing different parameters. We are using this tau c, that's the speckle optical relation time, and t, which is this exposure time. So start fixing ideas because in the experiment that you are going to see this afternoon, you need to fix the exposure time. And I'll explain you later on why. So you are having here a very interesting application of this dynamic speckle in biomedicine. And this is just now, is a very well known technique by the name of flowmetry. And what it is possible to study is the blood flow. So what you are having here, this is a healthy hand, the palm that is being illuminated just to create a speckle. And then having the procedure for digitally, you need to obtain this ratio here. And then you are just obtaining like a map. This is what it is named, the laser speckle contrast image. And it is named perfusion map because what you are seeing here is how is the blood flow in the palm of the hand. The blue part, it's a slow velocity. And the red is the fast velocity. So here, we need to introduce as well the parameter of the velocity that it's inverse to the contrast. So in that case, the more general approach for the contrast in this dynamic speckle case is the formula that you are having here. And we can consider be an ideal case of a Gaussian distribution for the velocity or a Lorentzian distribution for the velocity. And you see that in the case of the Gaussian, it can saturate and go into one. In the Lorentzian, we are not having that. But we can go to start from these values. This is represented. This is tau z over the exposure time. And what you are having here is the contrast. So as I was explaining to you before, the contrast is moving in between 0 and 1. And when you are having certain value of the contrast that it's close to 0 or a little bit higher than 0, what it is giving to you is the information that the scatters are moving. Because you are going to obtain a blurring in the image. And when you are observing a blurring, this means that you are having dynamic speckle. And then these scatters are moving fast enough that you can perform the average out of all the speckle patterns. So this is the way you see that we can make kind of a difference between the static speckle and the dynamic speckle. And I would like to mention to you this is a very, very interesting result that it's been published quite recently by the group of more the guys again. And it's that they have revealed the formation of a speckle in incoherent optical spatial solitons. This is the experiment that they have assembled. Essentially, it's like a Maxender interferometer. And then in one arm, you are having ordinary polarization. And then in the other arm, notice you are having here the rotator and diffuser. And of course, because you need the formation of this incoherent spatial solitons, you need a nonlinear effect. Therefore, you need here to use a crystal. This is the location of the crystal just because of the photorefractic effect you are just going to get a nonlinear phenomenon in which these beams are self-trapping. And then they are forming this incoherent distribution of the spatial solitons. You are having here how is this structure when in terms of the power of the illumination source. And of course, here, as I was mentioning to you at the very beginning, if the diffuser is stationary, it reveals the presence of speckle. And if the diffuser is rotating, you see that it's just behaving as a thermal source. It turns out that for what they are saying, I wanted just to show you this because this is quite a different phenomenon just to study speckle. And for what are they saying these authors, it can be used to study some surface, solid surface, material surfaces. So let us go now to some applications of speckling biomedicine is what it is named biospeckle. You can see here in this skin the possible different phenomena that we can take into account when light is interacting with a material medium. You see that there are many possibilities. And we are having this input. Then of course, depending on the reflection of the surface, we are having this speckle reflection. And for that, we have the Fresnel formula just to calculate the reflectance. We are having, let us take the direct transmission. This one, this is what it is called the ballistic photon. We can have a kind of a diffuse transmission here. As you are seeing here, this is a snake photon, the diffusive photon. And in this case, when light is transmitted, of course, we can use the Lambert law and also the Fresnel formula. And when we are having this diffuse reflection, of course, we can have absorption. Since this is a scattering medium, of course, we are also having a scattering. So see the amount of phenomenon that you can deal when the light is interacting with a material medium, and in which, for example, assume now that this is a biological medium. So you can have reflection, transmission, scattering, and explaining here. How can we use this phenomenon? So I'll take the example of the skin. You have here this skin of the dermis. It's the dermis and the epidermis. And then, of course, you understand that this is quite a very complex structure. We don't have to forget that we are illuminating. We are illuminating with highly coherent laser source. And therefore, all these structures here, they're causing effects as the one we were mentioning. You can have absorption. You can have partial reflection, partial transmission. And of course, you're going to have scattering. And this is because of all these inclusions that we are mentioning. We notice these epidermis techniques. It's going between 0.1 to 0.3 millimeters. And the dermis thickness goes from 1 to 3 millimeters. So in this thickness, when you can have all these complex phenomenon, and what we want is just to use the speckle. And the speckle, for the same example as I was giving to you before, this is the contrast imaging, laser speckle contrast imaging. And we also can understand that because the main component is water, we are going to have this is the absorption spectrum of the water we have here at the peak. So depending on which wavelength are we using for this process, we are going to have more or less absorption. You are having here the skin absorption coefficient and the skin scattering coefficient. So what it's possible to apply, as I was mentioning to you, this flow metering. In this case, what you are having here is the case of a patient that was having kind of a stain here in the face, and this stain it was treated. I cannot give you more details about it, but this stain it in the face was treated. So what there was, they were starting the dynamic speckle of the face before and after the treatment. And what it is observed is that there is an important change in what it is obtained. This is before. So you are seeing here the kind of distribution of this blue velocity. And after this is after 15 minutes of laser therapy, this is being more homogenized. So you see here a very nice example of a non-invasive treatment based upon speckle for these biomedical applications. So let us finish with the scheme of the experiment. You are going to see this afternoon the groups will be there in the forthcoming days, too. So what you are going to observe, this has been assembled by Professor Humberto Cabrera. So what you are going to observe is an example of dynamic speckle. This is the setup. So you notice, as I was mentioning to you before, you see that this experimental setup, you do not need quite a heavy infrastructure. You have the helium neon lasers here. This is the mirror, the lens. Here is the diffuser, the diaphragm. Here you are having the sample plate in which you are going to introduce the sample. And then it goes to the camera. And what you are going to take is a sequence of speckle patterns, a sequence. Because for what I'm going to explain you now, this speckle pattern is evolving in time. It is not a static speckle. This is an example of what you could use. Observe what you are going to use in this particular experiment in the lab. This is an ethanol drop. An ethanol drop has the facility that you can see very nicely the temporal evolution of the speckle. And it could also give you an idea that imagine that what you are having here is a biological sample with some cultive of cells. And then you can use also the same idea just to obtain this sequence of speckles just to see the evolution and temporal evolution of the structure of the sample. So we were mentioning that this is an ethanol drop. You have the camera. You are going to obtain a sequence frame by frame in the image processing. And you have to fix the detection time. Remember when we were just telling about the contrast in the dynamic speckle that we need to take into account the exposure time. So notice here this detection time or exposure time is the order of 15 seconds. And you are going to see now why. This is so short. Then you need to have a capture system like LabVIEW, the program made with MATLAB to compose these frame sequences, needing an algorithm and the method, temporal difference method. And then you can just with this arrive to see this activity of the biospeckle through the interpretation of the contrast as the contrast as a function of the number of frames. So let me show you now the video. So what you are having here is the evaporation process of an ethanol droplet. You are not seeing here the, oh, we see this is 24, 25, 26 seconds, 28. So the speed of evaporation of the ethanol drop, this goes very fast. And what it's important to notice, I want to repeat it. So what you may notice is that to study, you are not seeing here a speckle, because this is just to see this evaporation time. To study the speckle, you need an area somehow maintaining a little bit like uniform. So seeing that this is already 19 seconds, 20 seconds, 21, 22, so now it's collapsing in the surface. It's disappearing in the ethanol drop. And in saying that in 30 seconds, you are not going to be able to take any measure. So that's the reason why in this particular case, you need to fix this exposure time to 50 seconds. Of course, if you are just making a different application, you'll be forced as well to fix this exposure time. Where is the, here? OK, so this is the scheme of what you are, so you are going to see it in the lab. So as conclusions, we have introduced this concept of speckle phenomenon. We have been studying in a very basical way the formalism, the possible applications, the parameters needed, and then what is very important is to notice that right now, these days, this speckle phenomenon is having quite a lot of applications. I was mentioning biomedicine because this is the topic of this college, but it's also been applied in metrology, analysis of surfaces, and many other type of applications. And one of the things that I would like just to enhance for those of you who are just teaching, that this is a very good example just to introduce to the level of teaching laboratory because it's having a very, very nice content is to deal with the concept of coherence. This is very convenient to deal with the concept of coherence. And I'll mention to you these main seminal references, these two books from Chris Dainty and Joe Goodman. And also I recommend those of you who are interested in these biomedical applications, this article from Boss, it's very, very nicely done. And that's all. Many thanks.