 Okay, good morning everybody Today I will be chairing this morning session and I'd like to introduce and invite To listen the second part of be imagine and processing by professor Tatiana Aleva So good morning Today I will split the the topics in two So one of them will be a continuation of the yesterday talk about The problems of the quantity by imaging with white field microscopy and another one will be dedicated to The design of some special trapping tools in order to transport the multiple Particles of the micro in nano size. So let us continue with what we have Discussed yesterday. So I will refresh a little bit some content So we are speaking about the biological images biological like a cellular and They are quite Okay, so The problem is that they have a very bad Absorption contrast but so and we can treat it like a face only object So the first guess was if we are able to reconstruct the face of our image Maybe we have to know something about the sample and so they were developed developed several techniques, which for for this progress Based on the iterative algorithms Interferometry and holography transport of an text in the equation and face based tomography to name a few of them. Maybe there is a This is the main category which are doing this phase retrieval from the intensity measurement because the intensity measurements is the only one that we can permit in optical Wavelength So we consider the different Algorithms and we said, okay, we can Manage to get the face Sometimes when we consider the iterative algorithms We recover the face of the illumination beam together with a sketch and beam if we consider the holographic Pictures so we try to recover only the skater and beam but anyhow, this is the beam which is Which corresponds to the image and not to what we have Skater it before entering to the Objective so but the objective has its own point spread function So in order to know something about the object itself and not about its image We have to get out from this Point spread function somehow so there are several also method to do it and for this purpose we have to consider firstly how we can resolve The Helmholtz equations in during the propagation through the sample and in that case we can Represent these Helmholtz equations, which by the way, this is the wave equations, but Wave equation, but in the case when we consider monochromatic wave and the scholar approximation So we can represent it in this form where where we have Something which is corresponds to the propagation in the omahenius medium when this is n Zero stand for the mounting medium where we put our Cellular for example, so in the cell for example, so in this case it's correspond to the water or it's Propa environment and here we have The modulation of this Refractive index which corresponds to our sample which we exactly want to measure so this perturbation of Refractive index, it's quite a small but our goal is to recover that from our measurements So it is what we call like the optical like optical potential So we first solve this equation in the homahenius medium and after that we consider the different approximation In order to get the information about the refractive index of the sample So there are also different Approximation to do this one of them is a paraxial Approximation when we consider the accumulation of the refractive index during the propagation through the cell for example No, it is a canal approximation and paraxial approximation when we consider the scattering it's in the small angles which is related by the way to the Approximation To the method when which we used to recover the phase information using the transport of intensity equations But if we want to go further if we don't want to use a paraxial approximation So we can think about how to resolve the entire Hempholtz equation with other different Approximation and one of them might be is a born approximation which considers a small perturbation or perturbation of the refractive index and then it considering that the field what we recover is equal to the The field of the illumination which is staying here and then we consider the different order of perturbations and in particular usually considered the first one which is related and jailed with a scattering of the light through this Object, so you see that the born approximation is linear with respect to complex field amplitude, but in other approximation is a written one when we don't think that this don't consider that this variation of the refractive index is too small, but we have to Our limitation of this method that we consider that the gradient of the refractive index are quite smooth So they are not very big. So in the first case as a born approximation We consider only a single scattering and so if we remember the Gabber picture Gabber holographic picture. So in that case, we don't consider the diffraction of The interference of the scattering a scattered beam with itself We only consider the interference of the scattering beam with the eliminated beam so this is the limitation of the born approximation and In the rate of a case So so we consider soon only the one act of a scattering while in the rate of approximation It is Not linear with respect to the perturbation and this is linear with respect to its face Which in general this face is a complex Function because it's also incorporate the amplitude information and the face So it's only represented as this exponential form and therefore it is multiplicative with respect to complex field amplitude and this method in general taken to account the multiple Skatering and that means that it takes into account the diffraction of the scattering Light with itself. So somehow it is more each comparing with a born approximation So we have considered yesterday How this approximation work and the different application of that in particular for the different type of the Tomography, this is a face tomography, which is based on the applications of the a canal approximation after that The the scientists go further so they consider ready the first one approximation taking into account the diffraction and There is the 2p digital Allographic microscopy by the way the inventor of this method they even found their own company and So it's the method is working quite quite well But after that we decided that it's probably a good idea to get out of the Disturbance so which is related to the propagation through the objective which is cutting the certain frequency, of course The frequency which cut by the objective. We never can recover it But what we can do we can to change the weight Of this frequency in order to it's like in the Whatever linear system, which have the point spread function you can Make the deconvolution if you know information about it. So this is the same Idea and so as The scientists applied it as for the coherent as for the partially coherent light that we discussed yesterday also, but as a beginning for the applications of the first born approximation and After that we have so so this is The 3d face optical transfer function which allure us if we To make this deconvolution depending on the degree of coherence of our Measuring the intense is a three-dimensional stack of the intensity distribution Now let us say let us speak about the rate of approximation so we have Till now to a different way to thinking about how the information about the refractive index is Incorporated in our stack of the intensity distribution because in the case of them When we started with the face retrieval we said, okay We say something about the accumulation of the optical path passing through the sample, but from the Born point of view when we when we used the first order approximation of the First order one approximation so we suppose that every scattering is Independent and it doesn't relate one with respect to another one and only the Interference of the skater Light from the every this catering is interference with what with the illumination with the background illumination So it seems that it is quite opposite point of view on the formation of our Of our image so let us see if the rate of approximation Help us to make the bridge between both this Limits, I would say so in that case we consider our complex field amplitude, which is The real amplitude and the face in the exponential manner. So this is stay for Logarithm natural algorithm of the amplitude which is positive of course and its face And now let us substitute this expression in the Helmholtz equation when we do it we obtained this modified equations Up till now we don't Make any approximation. So it is only we change the notations Now let us consider so and we is as you remember This is our goal. We want to recover the optical potential so let us First as we did also with the born approximation Resolve the problem of a more genius equation It means that when optical potential is equal to zero and then we obtain the first Zero-zero approximation, which is our Beam which is propagated without any sample and let us consider now this face C1 is a perturbation phase which is appear during the scattering when our lights Propagate through the through the through the object. So in that case when we Add this additional term in our equations No noticing that this one is The solution when we is equal zero So we obtain the equation for the first approximation and if you make a Small mathematics and to arrange it in the server This form we find that is very similar what we have head for the born approximation if we consider this like our new function and if we consider this is I new Optical potential so this is optical potential and plus these terms So we can rewrite the integral Presentation of the differential equations and this is this one so it is Very similar what we have for the born approximation But in born approximation, we have this part which is staying for the complex field amplitude adding as the first approximation If we cut this part and if we don't use this normalization of the input field But now we are staying in the first order rate of approximation And of course it is impossible to resolve directly as this equation because here you have also information about this So one that we want to recover So what we can do we can resolve this equation also in its first approximation So we suppose that the gradients of them of this See which is an exponential form of our complex field amplitude like we like we wrote it Is too small and therefore we can drop it What is the price of doing this so we suppose the slow changes of the refractive index on the scale of lambda? Which is the wavelength and therefore we obtained this Quite nice expression Which relate the born approximation with the read the one so we can say that in the first order rate of approximation and also with this cutting of this term We obtain that the field is equal to the our initial unscathed field And as the exponential we have the first born approximation term divided to the first Do the complex field amplitude for the unscathed field The nice thing of this approximation that it takes the multiple forward only Scattering so it is already Better than then the first to born approximation, but it's more difficult probably To use this approximation to calculate the sink And to recover the information about the Psycho potential because because this term a stand in the exponential So now let us look what is the connection between the economic equation and the first born Approximation you we are staying with the first order rate of approximation. So if we only use the first two term of the Of this Exponential Taylor Taylor series so we get exactly what we have for the first Order born approximation So it means that we have to suppose that perturbation are too weak in order to Take here only the first two terms of the Taylor series on the other case If we consider that the scattering angle is too small Where here L is a distance of propagation and S is a perturbation scale Of the fraction refractive index so in that case the rate of approximation is reduced to the a canal equation Approximation and therefore we have this phase Accumulation due to the optical path difference between the light passing through the object and the light passing apart so we see that probably the rate of approximation is more More clear because it's considered in two limited cases both Approximation is a a canal with the people are using when they considering The face so so this is there are a lot of papers Which considers the recovering of the face and representing with the sickness of the of the cells and the Born first born approximation which is usually used for the diffraction optical tomography But of course, this is also approximation and so the people are Thinking about the following might be All of this Approximation is not sufficient to get the information about the optical potential Because the question is that we want to see some details which sometimes it's The scale is less that the wavelength scale why because we consider the Microscopic has a resolution the normal resolution is At least is the wavelength divided by by by two or even three so it depends on their numerical aperture that we have and so one of the Idea is to use the neural network to recover the information about the optical potential So one or there are several papers about this but This is in particular is from from it quite recent from the paper of Camille and this is a group of selfies and They decided to use this neural network in order to Recover the information about our sample so they present the sample which is in this case for example is a combination of a two-sphere is Is the different layers so in general they consider it like a box of 40 by 40 microns Which are divided to the Waxel and every Waxel is more or less 17 nanometers and they consider it 420 layers and After that they say okay how the light propagates through my sample so they consider that they have In every layer they in every Waxel they change the face Because they consider the face only object But they don't know exactly how to change it. So the first guess they are changing it Accordingly with the first one approximation that they obtained using the normal Optical diffraction tomography and after that they propagate the light from one layer to another one using the Fresnel diffraction and After that they compare what they have with their Directly with the hologram because this is the digital holographic microscopy method When they used also the different inclination so it is similar method that we discussed for them to be optical optical tomography and so they do it for 80 angles and and using the back propagation error algorithm after 100 iteration they recover their Optical potential and they have the following results. So this is the initial object So this is the field before the object. So this is what they use for this for for for the measurements This is the Reconstructions of the diffraction time when they use a born approximation for this method So you see that they have is that if we use directly this Optical diffraction tomography method Reconstructions, so you have here the information about the refractive index which doesn't exist Here where this representing these two balls in X and the direction it is prolongated which is quite known in general effect because because of the the famous missing call that we separate probably explain you Several days ago right and Here you have this in E y direction So what will happen when they use this loan algorithm? So you don't have any field before them The sample before this ball starting here They have the refractive index and X and this is in the direction So it seems that it's working, but it is of course very time-consuming because you have to Loan your system to recover it and Of course, you see that we move from the optics more and more to the digital processing and the digital processing sometimes working well, but sometimes it may be Like a trick and so we recover something, but maybe it is not wrong So this is a different Approaches that the people trying to to apply it in order to solve this complex Problem of the reconstruction of the three-dimensional information of our Images so something similar it was done also by Tian in Laura Waller so Not exactly with neural network, but also with some some iterative processes but using Not diffraction tomography not the holographic method, but another one so and there are also other Papers related to that So now Let us say a few words about the possibility How we can so we starting to think that there are a different method of the illumination So we're speaking about the coherent and partially coherent light and what is better and one say one aspect and another one So how we can quickly change one type of the illumination to the another one because of course you can do it moving your Closing and opening your diafrag in the back focal plane of the Of the your condenser, but it is not Not suitable because we usually want to get information very quickly So if you want to play with a coherent and partially coherent light to use this incline Illumination and that are so might be we have to think about a certain a certain Element which help us to do it so one of the proposal is is used DPL projector so that is something similar that we use it here, but it's more less It's not so so expensive probably like that one. So you can buy it maybe for 600 euro or even less and What is a good things of this projector? First of all, it has three Lets and so you can walk with a three different Illumination so that we read the green and blue then it is Incorporate the digital mirror devices you can buy the digital devices a part which are very very expensive But this for the illumination purpose is quite quite good The digital the mirror they don't have chromatic aberration which is also a good good things and they are have a very fast response in order to create the images and the Used before also for other type of the microscopy in particular for the structured illuminations and for contrast enhancement imaging and the last one the last one by the way, it is similar what is what is corresponds to the Coherence engineering because in this case for the country Contrast enhancement imaging what they use is as they put as a different filter. So the project using this project as a different Images to the back for complaint of the condenser. So it's exactly what we are doing but the difference is that in this application that they used for the normal absorption of the Images while now we can apply all this algorithm that we discussed before Recover the the quantity of information about the images. There are also alternative proposals for To fast changing of the kind of the illuminations, which is the lead era illumination which was used in this in particular in this papers So how We can do it. So you buy this is a projector you remove the lens which Project what we have here to the screen and you adapt your optics in order to project this Information that you have in your display, for example, it might be a circle jungle or whatever you want to the back focal plane of your Of your condenser and with that you can design your also So your image Can not to be binary. So it might be even you can design as a different type of the intensity distribution there you can to play with the different colors and therefore You can design the different Way to make the imaging playing with this Coherence degree of Diversity of your imaging formation which might be helpful for the reconstruction of the 3d objects so This so the images that I show you before in the beginning yesterday in the beginning of the talk when you See the ball the balls which are imaging and it's not clear what is corresponding to what if it is fear or no, it's correspond with this type of Application of this type of the system and as I told you you also have the the different type of Waiflets and so it is also useful in certain application for example in the method of the unwrapping or or to study the Dependence of the refractive index on the wavelength because we know that is a famous Koshir load that they are different In fact the Phoenix with respect to the way it's so Maybe I will go here so As a conclusions here you see the proposal of the Samsung and Blanca which used also the DLP projector to project the different pictures to the condenser aperture and they Tried to simulate the to to have the well-known a different style of Elimination that we know like a duck field which is almost that field that we have here like Hainberg Elimination when we do it with a different color of them Of the rings in order to separate the Different type of the law and higher frequencies and to see it's like like a optical staining and You also have to have the brick illumination. You also can To use this type illumination for their tomographic purpose if you put it like a different points in in time sequence, so you have this Oblique illumination which might be coherent if this this part will be too small and then to make the Optical diffraction tomography as we discussed before so there are a lot of opportunity using this Proposal for the coherence Engineering to get a lot of possibility to recover the image So finally what I want to say is that in general this Method which we discussed yesterday and today about the 2d and 3d quantitative imaging It is the realization of the ideas proposed in general in last century, but Realizable only now because we have the computational Facility that the people didn't have 50 years ago, so of course there is something something nuance, but mostly they Based on these On these old ideas and I want to say also to you that here I cited some papers, but There are a lot of them so it is very small Percentage that I did is so I want to apologize for the some after which have a very good work but we are not cited here and I want also to transmit to the following message in spite That there are a lot of things which have been done. There are also a lot of things that you can Did if you want because there are Several problems or no several many problems to solve because we want to fast data acquisition a fast data processing So we have to think we will have the law for the high frequency We have usually as a low signal noise ratio so we can think how to increase it We need the some rigorous method of reconstruction might be using again The neural network but with of course with some knowledge of the optics because otherwise it will not working at all We have to think about the proper sampling about the correct illumination We have to solve this right in problem, which is quite difficult if we all wanted to measure the sickness and Regularization method because we are speaking about the deconvolution deconvolution Mathematically, it's very simple. So you only have if you have the relation with in the shifting variant System between your signal and the point spread function of the your linear system So in the Fourier domain, it's only multiplication So it seems that it's nothing to do you divide it and you have already this thing But it's not the case because you when you divide it and when the Transfer function has the small value. So you enlarge the noise and so it doesn't work so there are a lot of regularization problems in that case and So finally, I would like to think my college or my my colleague who helped with this Rodrigo and Juan Soto who helped with certain figures, which is experimented figure obtained with our design Of course, I would like to say as a culture organizer with this is nice College that I Am here only for a few days, but I really enjoy it and our our Project from the Ministry of Economy and Competitivity that which for the financial support of some experimental results that I demonstrated here so And of course, I would like to send to you So I don't know if you want to ask some question now or we move to another part Which is related to the optical tracking? Thank you for very nice presentation good mathematics and some practical aspects Maybe anybody has a question to ask a first I can ask you Tatiana Can be these mathematics can be applied for describing metamaterials? for example These kinds of approximation can be used or not Maybe maybe yes, but it depends on the on what is the size because when you think about Metamaterials, so it's probably you are speaking about negative refraction index negative refractive index Maybe but I never I never seen it directly how they resolve it for this But I think yes, it's possible Any question? I mean it depends on what level you want to do it So it depends on the on the size of the wavelength with the details of your Metamaterials because it's consistent or there's some superposition of the small details So what exactly you want to get from that? Thank you for the nice presentation on your slide about the DMDs the use of a you said you you can use a projector Yes, a video projector for that, but actually when we had experience when using a projector We found out that are due to the pixel nature of the DMD itself. There are a lot of diffraction orders over there, so Using the video projector. Is that the same case here or it's totally different We put some diffuser So it's needed diffusers Nobody else, okay, we have enough time. I think we can make Let's continue the second yeah, so now we move to another part and this part is related to the optical beam configuration for the manipulation of micron and nanoparticles and This walk is mostly done Mostly related to the original research, so it was done with the collaboration of the doctor Rodrigo And so he include his name in this representation also So first of all we will speak about the light like an instrument for the small particle manipulation I saw that last week or next week you have some Representation also which is related with the optical tweezers So it is a picture that I have to speak before but Let us so I was because there's a basic things about that so the interaction between the light and which and in the particles it's a very known effect it was even thinking Almost 400 years ago by Kepler when he wrote his famous Famous book the committees when he realized that the light radiation pressure pushed the small object along the beam propagation Direction and they are deflect the comet tails which are consistent very small particles In the direction out from the Sun but the first laboratory But the first laboratory experiments on this stuff were made only at the beginning of the last century by Libidif Nikols and independently and It is in the same year by the way, so they demonstrate that exactly this radiation pressure forces exist But after the invention of laser the Ashken and his collaborators They have found that this radiation pressure can be used for optical manipulation of the Micro-particles and now also in nanoparticles and now it has a lot of applications and for Micro-nanoparticle control confinement respiration Cell surgery molecular motors atomic atom cooling etc So in the biomedical application, it's also very important So there are a single molecule studies like a motor proteins RNA and DNA mechanism interaction Then a protein interaction and if you want to learn more about that how it is working So I recommend you to to see in YouTube this lecture, which he has a link when I When you will see this presentation in the Winter college site so you can only the push this button and it will be redirected you to this to this lecture So it is a biological application of that. So in general what we used The optical tweezers used for this molecular study. It doesn't mean that you drop the molecule you drop the the particle which is the maybe the 500 Nanometers size which are touched to this molecule. So don't see that you don't think that you You make this manipulation with that particle and Also, there are a lot of stuff in the feed of single cell confinement For the measurement of the volume change and mechanical characterization of the cell cell surgeries, etc Also, it is important for cell transportation for their sorting And it is a good idea because you cannot contaminate Your sample because you don't touch them you used only the light for that and for ensembling organizing etc. So there are another Reviews that you can look of so probably as the first experiments which was done By asking and also his colleague on the optical manipulation into cell. It is this one of course, it's not look so nicely like the modern picture of the Trapping inside, but you see that this Particles a small particle they move from this point to that like you see in this inside of the of the cell and Is it moving like that, etc. So but of course, it's almost More than 30 years ago The people think about to make the micro-machines driving by light so they made there's some micron instruments like that and They engineering them in such a way then they can be Pushed to the rotation when they illuminated by the usual the normal Gaussian beam Because of the configuration of this motor, but can we do something different? Can we use for example the normal? Spherical balls and to move them with a special Light that we are prepared for this purpose. So this is something which is a contrary what we see here So the light is Gaussian, but the form of the particle is it such that it can be moved, but we may to try to To reverse the problem. So the light will be designed and the particle will be the simplest one So now you can even buy the optical tweezers. So there are at least two company Which is a very different approach to represent their products So one is from the store blood which is more scientific and it is more applicable And now let us speak about the optical forces So let's suppose that we have for the light with a scholar picture with a complex filled up to the describing by the amplitude in the face And there are two part of this scattering of these forces so one of them which Called usually like a scattering forces. It is proportional to the optical current or optical flux Which is the product of the intensity distribution of this point Multiplied by the gradient of the face. So it is this face This force which will help us to move the particle on the around Around the curve that we want to design in order to move them and to make something similar to the Micro machines that I showed you before but for the simplest particles and In other part of the forces are proportional to the intensity gradient This is a conservative of forces and they are Propulsion to the gradient of the intensity. So this force is Responsible for the trapping of the particle If we are speaking about the particles, so there are three regimes different depends on the size of the particle So there's a me regimes where the diameter is Much more than the wavelength in spite we are saying that it is much more in general. It's if it's more than 10 Waif lengths it's more or less okay to apply this This regime another one is a really really regime when the D it much smaller than lambda But of course how much smaller so usually if it's the smaller or More or less in the size of lambda we can say that it is also possible to apply and The intermediate regime is in general. It's more complex and they describe like Lawrence me model and If you want to learn more about that you can look to the following Paper which is cited here. So let us consider the simple thing that they consider the me regime so it can be treated in the Ray optics model. So you have The plane wave you have a particle and we look to two Rays B and A and if it is a plane wave so the light refract in the particle So we consider the particle with the refractive index which are larger than in the mounting medium And so as a ray Propagate in their direction they be read propagating their direction, but this is refraction all in them this ball provoke them The forces on the particle which are opposite To the to the force which Corresponding to the light refraction and the same happened in that case with with a ray A and therefore in that case you have only When you sum all these forces the only force that you have is in the direction of the Scheduling and so your particle will move in the direction But let us now consider if I don't have the plane wave and I have a Gaussian profile So again, we have two rays, but this raise B Has a less intensity. I would say so it's less intensive that a and therefore its force we will be More weak than the force that I have from the From the ray A and therefore we have the force which will Move the particle in the that direction and the direction of beam propagation But we also will have this a gradient forces We will move the particle in the region when the intensity distribution is larger So this is in general the idea of optical trapping. Why the particle trapped in the Level when we have the larger intensity distribution Sorry. Yes, it's not No, it's moving. Yes, but it's moving It's moving in both directions. So it's moving in this direction and in that, right? So it's still moving if we consider this This this beam, but it's moved in that direction and also in the direction of propagation, right? But now if we will focus it so the situation will change So if we will focus it so we will have the intensity gradients in the Direction, which will compensate this scattering force which is in the direction of propagation. So this is The of course we can so this is This is the simplest picture you have of course integrate over your these forces over all the The Object in this case is a sphere So in the case when we have the particles which have the refractive index which is lower Then the mounting may do so the situation is inverse and so your particle will move outside of the in in the direction contrary to the our intensity gradients Now if you want to trap the particles so as your colleague Mentioned so you need some stickers because your particles in general Propagate in the direction in the direction of beam propagation So you have to focus it and this focusing have to be too strong enough in order to It's probably it's probably later. We will see it Okay, let us look at it. So it has to be strong enough in order to compensate This is catering forces. So here you can also look to this To this website where you can play and how we can to focus them you can focus them using the high numerical aperture of your microscope so here you can find this Website where you can play with a different part of the Numerical aperture and to see how depending on that your particle can propel in the direction of the beam propagation Or can stop and can be trapped inside of this of this Optical trap so but now let us return a little bit back and let us consider The forces on the sub micron Relay particular a particle. So let us consider the electrical particles and we again can separate The forces to the two parts. So one of them is a gradient forces as I said you before which are proportional to the gradient of the intensity distribution and This also called like a lens Like property of the schedule because it's moving to the Direction of the gradient of the intensity it what is doing the lens when it's focusing it if the lens is Conversion or it's move it outside of that if the lens is diversion And this is depends on what depends on the difference of the Refractive index of the medium medium and the particles So on the other side as a scattering forces are proportional to the intensity distribution and gradient of the face and as you see here They are always are in the direction because you have a square here So they are always in the direction of the face gradient Here it's depends on if on the refractive index of the particle in the medium While in that case not that they always are in the direction of the face gradient We discussed it here. So what we what I want to discuss with you as how to move a particle So we see that it's very easy to move it in the direction of the propagation But may we can we make something? As a sink and we moved it in the transfer plane can we move it along the design trajectory in 3d How and how to do it? So first of all we can of course to trap the particle and move this trap but There are other possibility like a beam shaping so we will speak more about the beam shaping So we will speak about how to draw the two dimensional curve of arbitrarily form And what are the requirements for the particle confinement and transport along of this curve and What we understand on the optical Trapping How to propel I will show you how to propel the micron in nanoparticles along this Trajectory and introduce the concept of this polymorphic beam and Then we will speak about how to go from the two dimensional course to the three dimensional Which allow us to move the particle in Direction, which is a contrary to the beam propagation direction of course with what using this face gradient forces and After that we speak a little bit about how to create such type of the trap so if we We understand how we can trap the particle for example, which have the refractive index which is larger than the medium with the normal Gaussian trap, but it's usually it's not a Gaussian as a diffraction free Sport trap So if now we want and so we compensate the sketching forces in the direction of the beam propagation if now I move it I want to move it in a Certain direction in one plane what I have to do I can move the sport Temporarily so to move the sport temporarily. I can do using the hologram. So adding the Linear face we can move it in the direction move it along Along the 2d plane, but we also can add in the Spherical face and in that case we can move this point in that direction So it seems that it is quite easy to move the particles only Aiding something and Temporally and then to move this sport from one point to another Temporally related but but in that case we can do it with the Certain particles which are in the volume wet where In in the sport when it is a trap. So the people Did a very nice things for example they drop different Particles in their own trap. So you have to design for example if here you have Each particle you particles and you have to design their own movement of every particles and it will be quite complicated hologram Design and then you have to move it depending what you once every particle was separately but So it is nice decision Solution for some application, but after that the people think about but can we Do the single beam and so not a time Multiplexing it and to move and not might be the only one particle, but the several particles is sort of trajectory So the first idea was to use the optical vortex beam with the electrical phase And this is correspond to the Laguerre Gaussian elliptical beams Helical beams which Allow us to trap the particles along along this circular trajectory and and Move them Why we can move them because we have the transversal gradient which is Which is written here. So this gradient this parameter L is control how big is the phase gradient in the direction of Particle movement along this circle, but this trap existing because there are a lot of example of the similar Similar solutions and they try Different type of the particles or for example the particle with the refractive index with lower absorption Lower than the mounting that can be trapped but inside of the circle if the refractive index is larger They can interrupt in their direction, but the problem of Laguerre Gaussian beam, which is a Steel Gaussian that we cannot trap the particle Inside of the sample you need some help and this help is the cover slip of your of your chamber So this is a bad idea if we want to use this trap For example for the study of the dynamic of the particle, which is quite a complex process Because of course we have to take into account and that case the forces which are not only optical forces But the forces also of this glass that we have the super Superphase that we haven't said so the people Start thinking about how to create the normal there the proper three dimensional the true three dimensional only optical drop There were also other proposal to use other very famous Now and very popular beams like a iris beams which are moved along the parabolical trajectory And so it also was used for the optical trapping. So here you see some Image when the particles from this part of the chamber using this so this is the presentation of this I rebeam which was moved to another quadrant of these chambers But of course it is quite limited. So if we always will be looking for certain already known solution Maybe This solution is very limited. So why not to design? the beam for our Demand and not to use the word we already know from some solutions of the equations that the people Know already so What we what we are applying instead of using the known form of the beam to design it We is what we want, but what I requirement for Particle confinement and transport because we would like not only to trap you want to transport them And we want not to transport particle by particle. We want to transport it in Such a way that you can do it for the individual particle or for the Many particles. So first of all We need to design the bill the beam which intensity distribution following the arbitral Curve because maybe we want to move it Depending on the situation for example, we can surround the The cell or you can to transport it in the such a way that you're an obstacle So we want to design do to To learn how to design the beam which are following the arbitrarily curve in two-dimensional and might be in three-dimensional So, of course, we need a high-intensity gradients because because we want to create the real three-dimensional trap and not Use the help of the super face we also want to manipulate the signal and the multiple particles and We have we want to manipulate the object with a different optical property and my besides so nano micro particles and We want to control the speed how these particles are moving around this Designed it's core. So it might be uniform or might be not uniform. It might be bigger or less And it has to be independent on the form and the size of the core And of course, we want to create as I said you see dimensional trap And my bait to speak to think about this planning motion about the current situation detecting or avoiding the the obstacle so when we started thinking about that we found the nice papers of our Ramachkin and Volosnikov and They planning to construct the spiral beam. So the spiral beam is a beam which are Which have for the completely may have the completely two-dimensional the form of the two-dimensional curve and during the propagation they doesn't change their They don't change their shape, but only in large the scale during the simpler diffraction so We think okay, maybe it will be useful to use it for the trapping and Let us think how they are constructing this This beam in general we study to think about that because we have another proposal to construct it with a similar spiral beam But as a super Superposition of the Lager Gaussian beam, so it will be correct but it's not so easy to design the beam in the form that we are Wanting to have finally so the idea of Realization of the Abrabachkin and Volosnikov is the following so suppose that You want to design a beam in the form of the two-dimensional curve Which are written in the polar in the polar coordinate with the radius r and The angle t so what we have to do to design it so Is the complex field amplitude for the beam which will follow it this Curve is written in the form of the Superposition of the plane wave so Here you have x and y which is a coordinate of your final Coordinate of the your final complex field amplitude and Here you have a weight for this plane wave and this weight depends on that it's depends on the Modulus of the gradient of this of this curve And this term is say us that the intensity distribution along the corp is almost constant Which is a nice thing which we want also to have Then we have this term that in general Corresponds to the brush there's a form Of the profile that if we cut this corp So we will have the profile of your Intensity distribution and this profile will be Gaussian and case and the next term it Responsible for creation this is pearl characters So in that case your it is a guarantee for the stability of your spiral beam It means that the intensity distribution will not change during the propagation a patch from the scaling so In general There are a lot of possibility to construct them So we try for example you can have it like that like that like that and all of these beams with This is intensity distribution that this is the face will propagate it in a free space without change therefore So it is a nice thing. So like your Gaussian beam belongs of course to this class And it was also shown that it possible to drop them the particle and the move to Long this trajectory, but again against them the color sleep So what We propose to do first of all in so we are Thinking about the real 3d trap so we need the high transfers and axial intensity gradients and the Gaussian profiles will not give us Good Axial intensity gradients. So we have to change something in this formula And what we have changed we change this Gaussian shape to one because now we have To domain we have the representation of our signal in the domain, which is similar to what we have to the spiral beam, but the spiral beam is Is invariant apart from the rotation so in Fourier transform you will be exactly the same But in our case it will be not the case. So We will have the different picture in the Fourier domain and in the trapping domain. So Following the formula that I showed you before we change this Gaussian brush to the spot one which is correspond to the Substitution of the Gaussian from two to one and after that we need arbitrarily control of The phase gradient along the curve So before in this formula that I show you there Intense as a phase gradient along the curve was fixed and It was described by this formula why because the idea of this spiral beam is to preserve They are formed during the rotation But in general we don't need it because we will trap our particle in the only in one plane Right. So we substitute it to this form, which I will explain you later and And So all after after these changes, of course, we don't have them the spiral characters of the beam of the beam will lost but it will acquire other Good properties that we would like to explore so And this is a new expression for that. So you might be Looking for some people who are knowing something about the non-diffraction beam can say, okay, it is a very Similar to the non-diffraction beam and it is true Always when you get here the constant. So if I is a constant, you have a non-diffraction beam If are is dependent T It means that in the Fourier plane you Can to construct not only the ring but whatever trajectory which is described by by these are Depending on T formulas, you already have a diffraction beams, but which will Collapse to the tiny focusing Curve in the Fourier plane. So Now if we make the Fourier transform of this so using the our objective, so we will have this This expression which corresponds to the tiny core for which we predesigned by So there are a lot of possibility To design the core so one of them is to use the super formula with describe the shape of the plant and microorganism Which was proposed by gills when he started the form of the The beams which is described by these many parameters type So if you want you can try this one, but in general as a method that work for whatever To dimension of your life. So what now we get so if you consider how the light we are Gaussian beam for example It's profile in the focusing plane and compare the same Case for our type of a beam that in the case of The circular trajectory Transform to the best helical vessel beam. So you see that here is a very nice Transfer gradient which is represented here, but which is more important the very good axial Intensity gradient and this axial intensity gradient a low ass to drop the particle in Sight of the sample and not only Against the the color sleep. So It is a good things and if you want to know how we can design this type of the beams with the different Forms so this is the intensity distribution of their Representation in the hologram encoding plane when we go to the trapping plane they transform to the following Forms so this is a theoretical one. I don't know if you see them. So this is what you what you see in the upper line So this is a theoretical prediction and this is what you have managed. So we have for the very nicely defined the spiral the floor Flow wire and etc. And below you have the phase distribution along them now Let us speak about this is a real 3d optical trap the first real 3d optical trap or was experimented demonstrates in 2007 and by Rochman agree and they used for that the helical base in beam, which is a particular solution of our general formula that we propose and Indeed they will be able to trap and to move the particles along the circle and the inside of the sample But What we can do we can Maybe I will go to the To the formal. So this is the explanation how we can how we can model The propelling forces the forces that allow us to move with a different Velocity our particle along the curves. So they are related to this with this C SST term and the which will be Uniform if this condition is satisfied and non uniform for example this on whatever you want if it's if they are If you don't want it and here you see the representation of our Beams intensity distribution and the phase distribution is a different domain. So this is the domain where we design them. So this is the upper line and the Face face of them and on them on this line You have their representation in the trapping domain and here you have the different Distribution of the face. So here is the first column corresponds to the helical Basel beam, which is very known and this one is also ring But the difference with this one that the face gradient along of this is not uniform If it is not uniform, you see that it is also reflected in the form of of its Fourier transform of its hologram You can of course make the same intensity gradient by the rate The same the same face gradient along the curve but where is the intensity gradient that also will change them the forces which Move the particle along along that So it's correspond to these two part it and of course you can then play and to to make a lot of Different possibility to do that. So this is against the experimental demonstration of the Representation of this beam and now let us look how it is working and We This this picture corresponds to the trapping forces on the On the silica one micron size spherical particles Moved along the 2d trajectory. So this is a square. This is this is a jungle This is also this this is a normal ring and We can play with the different faces and so it depends on the global Accumulation of the face along along the curve the particle will move Faster or lower Depend on that and you also so here in this in this column You have the time lapse during the certain period time to see that exactly your particle following the trajectory that we redesign now we can think about the other possibility can we also Trap the nanoparticles and we are speaking about the metallical nanoparticles the metallical nanoparticles describe in the Depot Approximation and they also have these two forces one which is proportional to the intensity gradient and Which is related to the real part of the particle Parallelizability and another one which is depends on the face gradient and is responsible to the imaginary part of the particle Parallelizability and Again like in the case of the electrical particles the This force is Always in the direction of the gradient of the fee while this in general depends on the side or Alpha prime which can be a positive or negative and often it is too too low In order to trap the particles so it is the case in particular when we consider the so this is a picture for the Golden nanoparticles fear of the diameter of 18 nanometers and you see that Alpha prime can be negative If we are working with this wavelength, so we So it means that you cannot trap this particle using only the intensity gradient forces But what we can do we can use again The face gradient forces to trap this particle and also to move them But in that case we cannot create the real three-dimensional trap and what we do in these cases re-creating these traps against Against the cover sleep as it was also done for the case of the nanoparticles in the very primitive Trajectory it's again like in the ring or in the line where the simple Simple Conversion cylindrical lenses were used but in that case they could use The gradient in this is a gradient forces to maintain the particle. So What we use here focusing they beam we obtain some channels near This focusing where the nanoparticles are moved Inside and after that they moved along the trajectory in this plane So you can see it Here for the two parts of the nanoparticles or one of them Spherical and They are Gold and another one the triangle and they are silver particles so we can also design the The trajectory according if we have some obstacles as I said before using for them the spline line combination and Here you can see the the laser traps behavior for the case of or for for the case when When the trajectory after that is changing according to what we want to obtain and And the other picture you can see also You can see also when the stability of the strap and how it is How it is Maybe maybe you will see it after that how it is stable how it is can now Impact with some obstacles and after that you can move it Together all these Circles in that case is a circular trajectory So if we going to the part which is related to the How we can construct the real three-dimensional trap so we have to go again to the micro particles micro size particles which with the refractive index which is larger than the environment and Here you have also the sound some previous walks Also the group of the grill which show that the particle can move along the spiral trajectory in general It's only the half of the spiral and what is important that the particle can return back in the direction Against the direction of beam propagation. So they also show that it possible to do to this Two rings so there are two different type of the particle moving and we also can do it how we can do it We only if we consider that our 2d corp which it is all before it is a projection of our three-dimensional Corp and then playing with a certain type of the de-focusing we are able to Create the course which you can can see in this direction and so we check that it's Again, experimentally realizable so we can measure the intensity distribution We can move the particles The polystyrene particles of the size more or less of the five Microns so it's more or less like a me regime in this trap and they will follow this Trajectory like a spiral like incline course etc so After that you if you like you can try it, but maybe the more interesting thing into look how we are it is Going we when we consider it the one micron silicon nanoparticles. So you have such a wave A trap and they are going up and down which is a time-lapse you can see here and Finally you can see also what is going on in this in this case with these particles and if you see there is a quite complex combinations of this movement because there is some Bunching of the particle in some moment after that so it is a very rich dynamics particle dynamics that the people can study Using such type of a tree a trap and considering that they are created only by the optical forces without any any Any one uncontrollable one so Maybe I will stop in this point and how to create that you need the computer generated holograms some recipe and some Papers that I think are quite important to understand how to create this holograms this holograms using only face only face Only face Special light modulators is written in this arizone paper in 2008 and So if you are interested I can also speak with you and to make some recommendations and finally In conclude so this type of what we call the polymorphic beam because it can to change its form are important and once In the direction of the transport of macro in nanoparticles an application for the sales case for therapy When you can move the hot nanoparticles along a certain trajectory which we can surround for example There's a cell for drug delivery Microbiology on multiparticle dynamics study etc. So thank you for your attention. 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