 So today I will speak about Dectoral Tychography. So as you say, I'm a guy more in a spectrum of the range of the optics. So for us in optics, when we speak about images, we have a great advantage, which is that we have this microscope objective, which is a great component. So when we are dealing with images, what we do is that we just use this kind of powerful lens to form an image in the plane, which is called conjugated. And we know in textbooks that we have a resolution which is related to the numerical aperture. And what is important to understand is that in this conjugated plane here, we have an intensity. We have a light field of light, which is actually a magnified field, which is similar to what we have here in the plane of the object. But since we are only measuring the intensity, we have many the amplitude information about the light and we lose the phase information. So thus so far, I mean, microscopy had a great success for imaging. But what is sure is that it has this weakness, which is this inability to image the phase. So this is basically what was interesting for me when I started to hear about Tychography is that this ability to actually get the phase information that could actually be extended, of course, to the optical range. This sensitivity to phase is very regulatory for different kind of phase objects. For instance, specimens, which are more or less transparent, would give a very poor contrast, but also focusing components such or light deflectors and even more objects, which we call holograms. So there are many approaches that allow to image the phase in optical regime. So there are tricks. One of these tricks is so-called the phase shifting interferometry, which is also called digital holography. I mean, this is a big family of systems. And basically the idea to retrieve this phase is to make an interference between the field that we're interested in with the reference one. And here we on the image plane, we get an interference pattern in such a way that the intensity that we measure encode the phase information. So there are many approach that use this to get the phase information. Another one I would like to mention is the weight front sensors that you have illustrate here. So it's a component that is able to probe actually what was the weight front and how distorted it was. And there are many interesting applications that have been even in the biology with this kind of device, which is, I have to say, more adapted where the phase modulation is very weak compared to what we present today. And as you see, these techniques need references, which could be a very important drawback. And I have to say that these techniques also gets very complicated if it deals with polarization. So this is basically a very short introduction. The outline of my talk will be the following. First, I will give you a very short introduction to typography because I guess the audience here is very familiar to this approach, but I will spend more time to speak about polarization. And then I will show you how to imagine a kind of challenging specimens, which are natural one, which are the bio minerals or artificial one, which are metasurfaces. I will also show you that it could be also used to image light. And to finish, I will give you a short discussion for you to understand in which situation it's important to go for vectorial or if it could be doable to remain scalar. So, typography, as you know, I mean, it is a technique that allows to make images of objects. It's a computational microscopy technique. So you use this coherent illumination. What is important is that you need to be able to model this as with a multiplicative model, which means that you cannot have multiple scattering. So I mentioned this because for us in the optical regime, it could be a very strong limitation because some samples scattering a lot. And that could be a limitation for applying typography. So under this assumption, you could model the exit field as a multiplication of the illumination times the objects. And what you record is, as you know, the far field intensity pattern that you only record in intensity under good conditions, which means a scanning and an appropriate overlap. It has been shown that you could reconstruct the objects in phase and amplitude and the probe that has been used to probe this object. So here I insist on the fact that when you look at the retrieved maps here, so we have this probe on the left and you have the object on the right. So both of these maps are a map, which I call scalar because at every point of the map, I have a single complex number. And this is why I use this color map to plot the value of this complex number everywhere here on the plane. So that brings me to the point where I will speak about the polarization of light. So to imagine to understand what is the polarization, you need to remember that you have seen in textbooks that when you want to solve the Maxwell equations, you get that you could have a wave. And that wave is due to a field that oscillate in the transverse plane. So here this is a picture of an oscillating field and you have to imagine that this is a wave that is propagating perpendicular to the display. The general solution is the one that I've plotted here. So it looks so it's actually the red arrow that corresponds to the electric field that is a solution. And this is oscillating at a given frequency. And so you see that he described a nice ellipse. So it's a general solution and this ellipse you can decompose it into as the sum of an oscillation of the field in the X component, for instance, and the oscillation of the field at the same frequency along the other direction, which is the EY. A nice way to describe this is not to describe this real quantity, but to switch to this complex amplitudes that you see here. So you have here complex numbers and you took apart here the time dependence and the dependence along the propagation distance. So the idea that you have this state of polarization, you have different ways to actually represent this to model this so to quantify this. So one approach is the famous Stokes vector. So this is this vector which is made of four real quantities and which are calculated with the definition I give you here. So it's something which is very used in imaging because it's real quantities so it has definitely been designed for measurement. And what you see here on the definition of this and you see this time average and space average also about this EX and EY components that tells you that this property of polarization, it's linked to a coherence between the EX and the EY components. So for you who are maybe familiar with the notion of coherence, which is usually a phase relationship that you can have between the light at different places along the beam or transfers to the beam. Here it's a matter of coherence between the X and the Y component of the beam at a given place. So another way to define this property of polarization is the so-called Jones vector, which is simply the two complex quantities that you have here in the description of the field. So these are complex quantities. I have to mention that this description is only valid for totally polarized light, meaning that I need to have a clear phase relationship between these two components. But this description compared to the previous one has a great advantage, it contains the information about the overall phase. So to illustrate what I'm saying is just to say that actually the Jones vector tells you about the state of polarization and exactly where you are at a given time, whereas the Stokes vector only tells you about the state of polarization, but you have no information about the phase of the field at that time and at that location. So this is why in the rest of this talk, I will stick to the Jones vector description of light. So when you have this illumination, which is then polarized, I need to describe it with a Jones vector. So this is my illumination here on the left. Here I choose a linear polarization. And when this will interact with an object, general, I could have an exit field that could have a different state of polarization. Here I put a elliptical polarization. So if I want to describe the properties of my objects, then if since we have a multiple assumption here, it means that my object need to be described by the so-called Jones matrix, which is now four complex terms to describe every point of my object. So I ask you just to remember that the light is described by a vector two quantities, whereas the object is defined by a matrix four complex quantities. So now we have a Jones matrix that encompass all the optical properties you could have in the object, meaning, of course, the anisotropy related quantities, which could be retardants, linear or circular decrysms. So this is the ability to absorb light depending on its state of polarization. You could also have weird phenomena like optical rotatory, porous, etc. And it's also encompass isotropy related quantities, which are basically the absorption and the phase shift induced by an object. So we have established a couple of years ago that we have proposed a way to actually include this in the typography scheme of measurements. It means that instead of having one map to reproduce the object, we need to find four complex quantities for this. So the Jones matrix, and so this could be done by using set combinations of polarization. So it means that we need to introduce here the linear polarization of the probe together with the linear analysis of the output field. So we have several combinations here. So using nine combinations, we record a much higher quantity of data and using an iterative vectorial tachographic algorithm. We have shown that we're able to reconstruct the objects, which is so these four maps of complex quantities together with the eliminations. So we have here three sets of eliminations. Technically, this is made so we are in the microscopy in optical microscopy. So this is made in a simple inverted microscope. We have here an elimination on the top. The object is placed here. So the expensive part is here this scanning stage. We have here the microscope objective, which is forming here an image. And this is not the place here where we put the camera. We put here a field stop. So small aperture and we put the camera further at the distance about 20 centimeters to have the far field intensity, which is recorded for the tachographic scan. Of course, here we have a couple of components to control the polarization of the light before the objects and after the microscope objective. So this same setup is relatively simple. Now, just a couple of words about the algorithm. So we start like classical tachography with a guess. So the guess here is just that we have no object, meaning that it's like a diagonal matrix. So the first term is one here you have a map of zero, zero and one. We have a criterion that we need to minimize. So we try to minimize the difference between the square of the antenna, the square root of the intensity as it is calculated with respect to the one which is measured. And so there is an iterative algorithm where we do the propagation, calculate the far field, introduce here the information coming from the experimental intensity, inverse the propagation and update with appropriate gradient here. And so this will produce the results. Okay, so now I have described this method. I will show you what it gives on the rather challenging specimens. So the first I would like to describe is the one that we have studied in a group for years now, which are in the topic of biomineralization. So this is a piece of the oyster pink tada Margaritifera. So it has, it is very challenging in optical microscopy, because it's a rather challenging object. It is not very flat is rather thick, by the way, and it's made of prisms that you see here. It's something like have a size of the five to 10 microns. It is not completely homogeneous. And more importantly, and so that's really interesting for it's an isotropic. So this is what you see if you place that object between crossed horizons. And basically you have to interpret this image as where it's black. It's mean that the polarization doesn't change. The light was blocked. And this color, it's mean that you have a state of polarization that was changed by passing through these prisms. So here there are prisms of calcite. So we have performed pictorial technology measurement on this kind of sample. We have obtained these Jones maps. So we see these four maps here that correspond to the solution. And for the next slides, I will focus here on the rectangle which is here to show you what kind of structural information we could have out of this Jones matrix. So first, what we could have is the map of returns, which is the one that we would have seen if we have put this in a normal transmission microscope. We have here the map of the optical path things. So it's related to the thickness of the prisms and where you could see that from the left to the right, there is a gradient of thickness and actually every prisms behave like a small bump that we could hear monitor. And it's important to notice at that point that neighbor prism have very similar thicknesses. So you will see that it's important for the rest for the next slides. Here we have an isotropy related informations. And this is the fast axis orientation that tells us about the orientation of the crystal axis within the plane and the retardance that tells us more or less how much the crystal is an isotropic. This is kind of information we could analyze the more deeper way because here, as I told you, we have the two kind of information I to go back to the slides here. So what is very important here is that we have together something which is an isotropic information about the thickness of the object where we also have an isotropy information about the crystal itself. So that means that here when we see, for instance, that we have crystal of very different retardance, we know from the previous measurements that it's not due to different of thickness, but it's only due to different of axis orientation. Okay, so a cartoon picture that I could have is that these two have a different retardance, but it's not because one is thicker than the other one. The more appropriate interpretation is just that one has its axis more tilted with respect to the other one. Okay, so this is very valuable information that we could have that show us that depending on the present we are observing, we could have information about this C axis or it is oriented in space. Okay, and numerically, this is made this way. So we have the optical path length that is given by this index time the thickness. So knowing the index, we could get the thickness and knowing the thickness out of the retardance here, we could get the birefringence. Okay, in calcite, so it's a bit technical, but I have to do it quite quickly. In calcite, the index is defined by this ellipsoid of the indices in 3D. So having this effective retardance here tells us about how the crystal is oriented in space with respect to light with this kind of dependence here. So that we were able then on this kind of crystal to build these maps where you have here a color that tells you how the axis is tilted with respect to the plane of the sample. And if I look at front and these two crystals here, I see that this one is very homogeneous. This one was not homogeneous in terms of retardance, but this is just because the C axis was slightly, I mean the tilt of the axis was slightly changing in the cross section of the crystal that you can see here. So just a couple of words about other analysis that could be done. Knowing the Jones matrix helped us to calculate the Eigen polarization of the material. And this tells us, this Eigen polarization tells us what kind of polarization are able to propagate without be changed within the crystal. And there's a very known property, which is that if this Eigen polarization are linear, which has the lower left one here, it means that I can describe my object like a single bifurcation material. Whereas if my Eigen polarization are ellipse, it means that at least I have a super imposition of several crystals that are misaligned with respect to the others. So more or less, it means that if I look at this Eigen polarization at every point of the prisms, in positions where I don't have lines, where I see ellipses, I can tell. So this is the case here in this part of the prism, this part of the prism that I have singularities in the structure of the crystals that defect in depth. Okay, so this is a kind of very detailed structural information that is provided because we had the possibility to have together the phase and the retardance properties of the crystal. Okay, now I will move to more artificial objects, which we called metasurfaces. So metasurfaces are devices, components, optical components that are based on the structuring of a material at a scale which is smaller than the wavelength. So you see here the kind of material we're studying. This is a collaboration with a group of Patrice Genvay in Valbonne, who is an expert in the study of these metasurfaces. And so that kind of objects have a global response to light and the way they are manufactured or the way they are oriented, their thickness, et cetera, introduce specific phase shift or change of state of polarization. So that means that this kind of small object for us in optics are quite fascinating because this is kind of an optical toolbox, meaning that you could really tailor the optical property of any components. So here I will show you a couple of examples, but typically the kind of thing that could be done is typically a component where you illuminate that component and display you an intensity pattern here where every part of the pattern has an appropriate state of polarization. Here you have another one which is quite fascinating. This is an object. When you look on this surface you see a figure here with your face. And when you look in the far field, so on the holographic projected image, you see a pattern which has nothing to do. And all this is made on appropriate polarization. This is also an example of how to use this to create weird beams of very weird special distributions. So imagine a meta-surface. It's a challenge in terms of optics because we have to map quantitatively both phase and polarization property. So as you have understood, this is something which is for which vector ultra-graphy is very appropriate. So the sample, I mean one of the interesting sample that we have studied is the most recent that we have made with this group is a pattern in such a way that when you illuminate this pattern, with horizontally polarized light, you get an intensity that is homogeneous. So you expect to have projected intensity that will be a square of light, whereas actually the polarization property within this square will have different patterns, meaning that if I plot a map of the orientation, I should see this shape whereas if I plot the ellipticity of my light, I should say this rocket here. So this is a sample that we have put on our setup. I show you here the Jones map. So more or less, so you see these four impressive maps. So it looks very structured. It is not very impressive because you have to keep in mind that this is a hologram. So if you look closer, it looks like this, but it's this kind of picture is not very informative. But to show you that what we have found is actually the real object, we could use the property of having this Jones matrix, meaning I told you that this, the great advantage of this Jones matrix that we have the possibility to know exactly the phase, the overall phase. So it means that if we use this, we have the property possibility to check how the field will actually propagate after that object. So this is what we have done. So from the reconstructed Jones matrix, we have made the following simulation. So we multiply by a given illumination, but we will get a resulting vectorial exit field that we fully propagate to a monitor, what will be the vectorial far field of the holographic image based on or measurement of the Jones matrix map of this metahologram. And so this is what we have done. So I show you the lights, how it's the way it propagates. So you see that depending on the distance which is displayed here, you see the square of intensity which is forming. And you see also that the map of orientation and ellipticity displayed the desired shape. Okay. Okay. So here I focused on the imaging objects, artificial or natural, but I would like also to show you that it's possible to imagine light fields and there are some topics for which it's a very critical. So now so far we had so far we had several illumination, and I have shown you that we could image any an isotropic or isotropic specimen here by mastering the polarization before and after. And for this kind of measurement that I have shown you before the only knowledge that we need was only the state of polarization of the illumination. So we only needed to know that it was actually linear with the angle that were desired. We have shown that it's actually possible to use an unknown illumination beam that you see here. And to analyze it, using the same vectorial tachyrography setup, and the only constraint that we have is to use here an object that could be that needs to be only non-barreffringence. So a piece of glass or anything and scattering. And with a non-barreffringent and scattering specimen, we could use any vectorial light field just by using three analyzes. Okay, so here if you compare here the picture here compared with before, before I had I retrieve an object and three illuminations. Here, what I will retrieve is the object and it's important to check that I get to retrieve the appropriate object that I knew before, and that's a guarantee that what I get here as an illumination field is correct. And so the illumination field that I get, as you remember, is a Jones vector, it's a map of Jones vector. So it's again these two components that will be EX and EY, map of each represented with this color code here. So the first one that we have obtained was the analysis of a radial polarization. So this is a kind of beam that you can do by using a so-called S plate. So you illuminate this with a circular, sorry, linear polarization and we get a beam within which the polarization is radial like this. So this is the kind of intensity, a map of field that we get for EX and EY. And here you have to understand that I have all the necessary information to understand how this beam behave. So it's not very easy to understand here when I just show you this very colorful image. So to show you more like what it looks like, it's better to actually make the field oscillate so you could really see here what kind of beam is this vectorial beam that you see here. And what you see here in particular is that it's not a perfect beam, so an experimental one, but you see any way that you have a radial polarization. You also have some regions in which there is a weak elliptical polarization. And there is also here a weak wavefront curvature. So you see that it's a quite precise tool for characterizing complex beams. Another kind of beam I would like to mention is this phase vertices. So which is also called angular momentum, orbital angular momentum. So here you have also again, so it's okay just tell you a couple of words about how it's made. So it may again with this S plate, but you have to eliminate it with a circular polarization and the output after this is a field which is quite difficult to understand. But just here, this is the wave, the phase front of this beam is twisted like this. Whereas here before, if I would have put the phase front, I would have put disc that propagates along this distance here. So here you see it's much more twisted. And here this is the map of the EX, the map of the EY. And again, I'll show you how it behaves. And so you see that we are able to describe the field in its entire property because we see different kind of things. First, we see that it's circularly polarized lights, as you can see everywhere here on the beam. And in addition, there is this orbital angular momentum, meaning that the phase is actually rotating. If you look at where the field is, let's say on the right, the portion of the beam where the arrow is on the right is actually rotating like this, which is the characteristics of this orbital angular momentum for a beam. It's also possible to analyze very completely arbitrary beams. Here this is a speckle that we have made by using a scattering field. So we got a speckle and the image of the speckle is like this. So we've seen this grain of light. And as expected, you see that the state of polarization is the same within any grain of light, which is something that was expected by the theory. And you see there is no phase correlation between grains. So that's the characteristics of a speckle. Okay, so now I just would like to say a couple of words because I have shown you many examples of that use actually this vectorial typography. So one question that may arise is when do I need to use vectorial? What are the conditions for which I need vectorial or can I remain scarred? So this is why I call this when does polarization matter. So if you want to find it to answer to this question, you have to know how does the investigated object affect the light polarization. So the question is not about the light you use, it's about how the lights polarization is affected. So that means that if you have an object, any object here that illuminate with a very specific light, which is here, for instance, an elliptical one. And that object generates, let's say an absorption, you see that the light is weaker here, also a phase shift. But if I keep here the same state of polarization, this light does not affect polarization. So that problem could remain treated as a scalar one. And so I can have this way of treating problems as a scalar for any polarization or even unpolarized light as long as the states of polarization remain the same where I measure. Okay, so but there are some situations also where polarization is affected. And here I'm just giving you some examples about resonant magnetic scattering because this is some calculation theoretical calculation that I've made a couple of years ago. And I think it could be interesting for some of the people in the audience here. So, don't ask me too much about magnetism because I'm not an expert but anyway I understood that's when you have in this situation you could describe your object by this kind of matrix here, we have some charge and magnetic scattering which plays at this position in a Jones matrix. So in this case, if you want, depending what you are interested in, if you are interested in the charge information, then of course here the answer is that you just have to target this term of the Jones matrix, meaning that you just have to use parallel polarization with a single measurement and you will get a map of the FC. If you are interested, on the other hand, on FMMZ, which is this one, then by observing this one tells you that you just need to use cross linear polarization and you will directly get this information here. Okay, but I mean this is very specific and for instance, if you want to use a circular polarization and no analysis and it has been made by this group here, you could very easily check that what you will be measure depending on the way you polarize if it is left or right, you will have this combination with a plus or a minus. And this is a way to actually measure properties of decrease of decrease of magnetic decrease and this has been made here in this way. Okay, so that show you that actually depending on what you are interested in. It is maybe not always necessary to go for Victorio. So depending what you want to have another example here which were recently in a literature where people were interested in linear decrease. So this is a case where you have an absorption that depends on the polarization. And here the approach was actually to use several angle of illumination, I mean angle of linear polarization illumination, so that was several scalar typography measurements and reconstructions. And so they could get different kind of images depending the way they actually illuminate their objects. So this is another way to use a kind of scalar typography even on materials that affects the polarization of it. Okay, so I'm done. Just to summarize this. So this is a computational phase retrieval imaging. What is important is what you image are Jones matrices. It offers a very reasonable resolution in terms of microscopy it's adapted to very large object. And it's suitable for precision dependent materials. And it has the advantage to provide a quantitative diagnosis for optical components, especially when they evolve phase and polarization together. So it could be a nice approach for reverse engineering of metasurfaces for instance. I've also shown you how to imagine a vectorial light field. A couple of words about my colleagues, I will especially thanks to Arthur Baroni because most of the things have presented here were done during his PhD. And now he's a postdoc at PSI. And to also my colleagues Mark Allen, Peng Lee, Peng is now at Diamonds, Julien Dubois and Virginie Shamar and King was song and patrician vein co worker at the crowd were the guy who actually made the metasurfaces and you for your attention. Thank you.