 third try Good afternoon So When when you have waves being emitted by different sources sometimes you you observe that these Waves interfere with each other sometimes you don't if we think of light sources for example, they have two Torches to lanterns and I point or the light from these two bulbs if I turn one off I see some light if I turn the other one off I see the light and when I put them together I just see twice as much light They're not they don't seem to be talking to each other don't seem to be causing interference Well, if I do the same thing with a laser I break it into two parts and then I join it I will see these interference fringes. So what is happening there? it's all about Statistics and and this concept of coherence is some averaging that happens and There's an expression English that says beauty is in the eye of the beholder that is beauties in your eyes Not in the thing that you're looking coherence is also like that Coherence is in the eye of the beholder what is depending on what your integration time is or what type of detector you have Something can be coherent or incoherent or something in between is not something intrinsic of the field is some combination of the field and How you measure it? So this agree what coherence is is it's about is if you have some source emitting some complicated field The coherence is a property between what the field is doing at one point Let's say and at another point or at one time at another time and essentially is a measure of how much Correlation there is between what's happening here and what's happening here how much? How much can you make those two things? If you know the field here, how much can you tell me about what's going on here? Now I like making this analogy suppose that these are water waves So we're somewhere to see it's nighttime. So you don't you cannot see and you're in your little boat you work for for For the Italian coast guard and you're in charge of being somewhere and measuring the waves So so you're in your boat, and then you feel like your boat's going up and down up and down and then You have your your your friends in another boat here Just a few meters away and and Then you know that when you're going up with the wave chances are your friend is also going up and Then when you're going down you're going down So you have a large correction you cannot see your friend because it's nighttime But you can call him on the phone and say yeah, I'm up. Yes me too. I'm down. You know me, too so there's perfect correlation or your friend could be let's say here and And then you say I'm up and friends are I'm down. I'm down. No, I am up Do you have correlation there or not? Yes, just as good it's anti-correlation is it has a negative sign But there's correlation or if you even if you're quarter of a wave, so I'm up Well, I'm on my way up and etc. So so as long as you can predict what's happening then then that's fine now if if then your friend starts their boat and they start moving not 10 meters away, but a hundred meters A thousand meters very far at some point you start losing that correlation because the waterways are so complicated that Over time when you average over time There is no much predictability of what's happening in one place from what's happening at the other place So that's what we call roughly speaking coherence. How much power of correlation you have between two points? and when It is in the direction of the propagation of ways the separation is what we call longitudinal coherence when it is in this direction is Transverse coherence and you can already see that they're this concept of coherence length and coherence width which is If I over time we're on the phone and we correlate our what we feel So when he's very Close of course, we have for this distance if I use myself as the origin we have a lot of correlation But as he keeps going away It's less less less less and at some point If I'm here and my friend is near Japan then there's no correlation so So we can define some coherence length as some distance where this drops to some Halfway level or something like that Similarly with the coherence width in this direction if I use myself of the as the as the reference You can keep moving here and you see that there's some some separation over which we have some some Given correlation so that is Basically the idea behind these coherence theory of coherence so when we have Source that is very very monochromatic very very deterministic then there's predictability over long long distances, but if not It takes all it then it is more limited The same with time I could have done the same thing here with say So I'm sitting here on my boat and I'm taking records I'm up at this time and down at this time above in this town and then I make a prediction Okay in 10 hours am I gonna be up or down or in five hours, etc And and again your predictions are gonna be pretty good for some time And then they're gonna start you're gonna be losing that and then you cannot predict beyond a certain time So that would be temporal coherence. How much correlation is between different times? Okay, so where does coherent comes from? This is a very simple cartoon of how For example a thermal light source would work so If I have some object and each atom or molecule is emitting Oh, well, we know that these have levels and we can excite it and then it the case and when it the case it emits light I'm using a classical picture here So I don't want to call this a photon because I don't want to get into what a photon means It's just some pulse of light and it has a given duration and And each one of these particles is going to be emitting like this several times and in addition There are many of them each one emitting they didn't meet more or less at random times So if I see this one is emitting then and then for a while doesn't do anything then it emits again and it meets again Then it meets again at some other random time same with the other Atom or molecules and with the other and when I add them all up at some detector I get some signal like this So what can you tell me about? the coherence time of this source Over what separation of times do I have some predictability? The coherence length the coherence time this is the time axis So the coherence time which is related to the length of the individual pulses Think of an analog of Picture say you have a road and you have a car here a red car here a White car, sorry Then I have a red car green then I have a blue car Brown car Etc. And then I I I Cannot see the whole photo. I just see this slice here So over how much distance can I predict what I'm going to see if I look at another slice of this? Only over the length of half the length of a car if I move by half a car Distance then I can predict what cars I'm going to see if I come here Have no clue because all these cars Have ended so so where the new cars what color they're going to be etc. I just don't know it's the same thing here But I have some power of predictability if I know the field here more or less for some time Around this Equivalent to the width of each one of these pulses because if I go far away The pulses that are given rise to this field here are completely different Than those ones given rise to the field here. Therefore. I have no Predictability, I just don't know if the face is going to be up or it's going to be down So this is what causes the the it's a way of understanding what causes this temporal coherence the fact that There's a randomness in when these things are admitted even if they were identical They're admitted at random times and therefore I can only tell you what the waves I can only predict what their waves are going to do for for some length of time Equivalent or related to the length of each one of these pulses That idea makes sense Yeah How can we measure that? Well, there's this classic experiment Where I can With the Michael's interferometer, so I send my light source and I have a half mirror here So half of the light goes this way bounces back here have goes this way about back here So I can interfere the light with itself Causing a delay by moving one of the mirrors so I can by moving the mirrors I can I can cause a delay and if the difference in path is very small Then I can see interference. So when it's very very small, they're in face They're constructive interferently each one of those pulses and I get a lot more light But then I move it a bit By half a cycle and then they cancel mostly then it's constructive again, etc But they start I start losing this as soon as I make the separation Bigger because the amount of light that comes from the same pulses that are inhabiting at the same time In this detector is a smaller and once my separation is longer than the typical length of a pulse There's no there's no correlation So this is a way to understand this concept of temporal coherence It's the time difference over which I can get Interference that is robust in time Let me stress that if these were an instantaneous Detector that just at one time measure the the light I always have the superposition of a field with another field and that would give me some crazy fringes But because this is has a detection time much longer than these wiggles of the fields Some at some times these things are going to interfere constructively as some of the ones They're going to structurally and they're gonna average out and I'm gonna get that this averages to something pretty boring Well for these small separations It's always in phase or always out of phase and I get this this This persistent fringes Okay That makes some sense Let me then I'm going fast Okay, I'm gonna skip that math. Well, let me just Show you quickly what this is. So so how would I define the field in the detector? I? Would have the field that went through one of those paths and then the field went through the other path and and this is delayed a little bit because it went a longer path and What we measure as Anna has told us several times is the intensity the modulus squared of the field So I have to take the module square, which is this times its complex conjugate Because it's something plus something times something plus something we expanded is the square of the first the square of the second and then the cluster and Then we these average this brackets here means we have to average in time because our detector is low is integrating in time so this is just the light coming from one path and if I average it in time is Fairly constant. It's like the light from this light bulb If you were to plot what the electric field at each point is doing at a very fast scale is doing a crazy oscillation But if I look at it looks pretty boring to me looks Constant because the average is constant My my eye is not fast enough to to see all these all these oscillations So it looks constant and this would be constant again Same with this. This is just to late a little bit of time But this cross term is what gives us the what we call the correlation So the correlation is the average in time of the field with a delay with the field with no delay It's not so important for this Course but what I show here is something called the Wiener Hinge in Theorem which tells me that this correlation between what the field is doing at one time and what it does at another time If I take the Fourier transform of that I Get the spectrum of the source So the Fourier transform of the spectrum of the source is not the signal in time is the correlation so a Source that is like a laser that has a spectrum that is very sharp Has what type of a spectrum? What type of correlation? so if the spectrum is Very sharp like this. So so this is as a function of frequency. This is the spectrum So it only it's only emitting at a very fixed frequency then the Fourier transform something very sharp is something Very wide. Yeah, so that's a very coherent field that the correlation goes for a long time Well Some source like this one has a much wider Spectrum so it's correlated over much shorter times How does the laser work? well It's very similar, but uh these little trains are Are Each one When it goes by it triggers another one So the second one let's say it's not it's not emitted at random It's emitted in coordination with the first one So they are in phase and then this one might emit another one etc. And when you put them together it gives this Something that has some correlation that is much longer than than the pole the duration of each poles Using my my analogy with cars Here I have a train track And I have a train So each car of the train is not unrelated to the others They all look the same and they're all fixed in the same distances So if I can see where the car of a train is here, I can predict for a long distance what they do So cars are like pulses that are each one at its own pace. Well, this one is they're tight they're they're they're linked together okay, so that's Temporal coherence. Let's talk about spatial coherence, which is uh more more interesting Suppose that I have some source here Source that is let's say made out of many particles that are emitting incoherently. So I have some waves like this and Suppose that I want to measure how correlated the field is For a given distance d How much correlation there is If I am at a distance capital D from the source and I find that There's some correlation say I choose this to be the point where the correlation drops to about half Or say I can separate this more and more and I can measure the coherence with Now I come further away And I repeat my experiment What do you think as you propagate away from the source? Does the The coherence width get Longer larger or smaller? Do we gain spatial coherence or do we lose spatial coherence? So who says lose Okay, so that's That's Sounds very intuitive. So that you would disorder Usually wins and you get more disorder as you propagate So You would think the further away I go the more the the more disorganized and the more separated I would have to make my uh, sorry that The less correlated this would be with this as opposed to this I'm sorry to tell you that is the other way around And that's what's called the van Ceter There's a mathematical description of this, but I want to give you more the physical intuition And I'm going to show you Something that is perhaps the shortest And most unusual and weirdest Publication I've ever had the luck to be involved with It's called spatial coherence from docs And this was published in In physics And what is it physics today? so So my co-authors are Wayne Knox who used to be the director of my center and emu wolf who who wrote some famous books about coherence And the story behind this very short that's the article by the way, that's it And that and those are the figures the docs The story behind this is Wayne Lives or well he moved recently, but he used to live In a house in the country And he had this nice pond next to his house where he kept his docs And one morning it was very quiet. So he couldn't resist bringing his uh camera and And filming When he goes and and frees his ducks. So So this sounds and everything So there's this early morning is very quiet To look at the surface of the water Okay, this is takes a while. So let me move forward. So that's where the ducks are. So he's walking there They're here the birds Soon the ducks are going to realize he's coming up To the house. They're going to get very excited about now and he lets them out And there they go directly to the water Now let me freeze this here This I mean there are many other effects here that are not a strategy here, but but just like uh at the very simplistic level Each dock now is an independent source Each one's kicking at its own rhythm is not worried about what the neighbors are doing and it's emitting waves Near the docks I'm getting waves in many directions at random things and there's a lot of the sort. Look at the water here If I let it go over time if I'm here, I don't know what the water here is going to do There's a lot of disorder because of of of this randomness of the docks wrong computer um If I let time go and then after a while I go and look at the edge of the of the pond Have this beautiful long wave friends That means that I can predict very well what the waves are doing Somewhere else compared to where I am And this is nothing but simple geometry Here are my docks and I'm going to show you some simulations in a minute Each one is emitting More or less a circular wave Points like this The way from this one and the way from this one and the way from this one are coming from very different directions so in time The the evolution in time is going to be very different from this contribution than this contribution than this other contribution And if I move a little bit All the the relative phases between them are going to change very very very very uh drastically So I just need to move a little bit and the the relative directions all the waves are very different If I come over here Keep going. This is the way from one dock This is the way from from another dock This is the way from from another dock They're extremely similar They they are doing pretty much the same and interfering in phase For a long long distance if I draw this is So this is the way from from one dock. Let me use color This is from another dock It will take a long distance laterally to have the contribution from one dock and the contribution from the other dock Go out of phase with each other It would take a long distance laterally Well here, I just move a little bit and that's it. So The field I feel here and the field I feel here is very very similar And that explains this band theory of certain equilibrium is that All that matters for coherence with is essentially The solid angle or the angle in this case Subtended by the source Because that tells you the range of directions in which you're getting waves If you're very close, this is a big angle that the source is obtaining But if you're far away Everything looks like a point source if you go sufficiently far away And if it's a point source is sending you something that is essentially a plane wave, which is very organized So that's why there's There appears to be more order As you move far away There's not it's not that you're getting more order is that The disorder Is spreading over bigger circles And therefore If you just look at a given region, you just see something that is fairly ordered That makes sense Okay, do you want to know the end of the docks? The sad part of the story The docks are dead, I'm sorry So they were eaten by foxes By eagles apparently And this is Wayne that tells me this I don't really believe him very much, but he claims That there are turtles big turtles in that that come Grab them and and eat them. So So there are no docks anymore But uh, but they contributed to science By the way, this video is on youtube if you want to if you want to see it I think the link is here All right very good so Let me show you how that works. So, um I have three movies here suppose that I am an excellent dog trainer And I told my dogs You stand there. You stand there. You stand there. You stand there. You stand there And then I tell them one two three and I start directing them and they all kick at the same time This would create, uh, a wave that Play this as a movie This would create a very coherent wave because there's no disorder and not only is it very coherent. It is very directional So the more order there is in the source The more directionality you can achieve. This is like a plane wave So this is what a laser is doing essentially instead of docks. They're they're molecules that are also being synchronized by themselves So it's not me telling them is each one is feeling the waves from the other one and and responding saying Oh, I like that rhythm and going the same way and and and then it would emit this thing So so this would be like a laser source Uh, that would send a very directional wave If I am a even better dog trainer I could do the following I can tell my dogs standing line Kick at the same rhythm, but you kick very little you kick a bit more a bit more a bit more And then I do this apodization that we were saying so I tell the dogs just Do your your like kick at a different strength each one of them And this would create something very similar To this calcium beam. This would really create a very very Coherent uh wave And that's how yeah again how a laser would work But uh Docks are not like that And a situation more similar Okay, mathematics freaking out Okay Let me stop this stop So this is more like what happens with the real dogs They're at random locations And I'm having them kick at slightly different rhythms each one of them slightly different frequencies and what we see is that There's some nodal lines where there's not much waves But it goes in all directions and those nodal lines actually move around as time goes by So if I were to make it to integrate in over a long time that intensity It would be like this lipo is going everywhere So to have directionality You need some measure of order of the waves uh, if not Your source is going to meet In all directions and that's the difference between a thermal wave A thermal source and a laser for example How would I measure a spatial coherence? So the there are many techniques, uh But the the classical one is if I have my source and I have some some wave To measure how correlated the field is here to here What I do is I put a a plate here And I just put two pinholes at those positions Now this light is going to hit here and it's going to emit A wave like this And this one's going to emit a wave like this And then I can measure That light And then I put a detector on this plane If I cover one of the holes, I just get light from this one I would get something fairly uniformly illuminated If I cover the other one, I would have something fairly uniformly illuminated But if I open the two I could or could not get interference What why wouldn't I get interference? Yes Is it's more than the coherence with yeah And again if I were able to take a snapshot An ultra-fast snapshot of what light is doing at a given instant I would get some interference Some some some highlighting, but if I did it the next instant It would be something completely different If I did the next instant it would be something completely different and when I integrate it It would average out to something boring However, if these points the field at these points is very correlated Whatever I get here at a given time Is very similar to what I get At another time and what I get another time and it would persist and will give me a persistent pattern Like this So it is about You need integration in time. That's why I say coherence is in the eye of the beholder. You need to Specify what your integration time is For example, we did some experiments With some students where we needed to prove something worked when you had some Incoherent superposition of two things and to change from one to the other we had to Change the orientation of a wavelength and Well, uh, if our integration time was in the order of minutes We could change it by hand back and forth. Let's say and that's To the detector that was incoherent because the integration time was much bigger than that So it doesn't have to be in a nanoscale So there are sort of Several length scales one is time scales one is the time scale of the oscillations of the field And the other one is the time scale of the detector And uh, those two very often are many many many orders of magnitude of difference And there there might be The time scale of what's of of the fluctuations of the field So let let let me illustrate this uh also with some simulations here. So this is a very cartoony version of The this is uh, let me do the previous one So this is a cartoon version of what happens with a young pinhole experiment I have a point source here Have two pinholes. I have a plate with two pinholes And this is the interference that I would see with a point source monochromatic point source So I can move my source to the sides and I can see that the maximum and minima would move because the line Going through the center of the two pinholes has to go to the maximum because this path and this path Have the same length and then we have constructive interference I can also I can change the separation between the pinholes. I can bring the plate a bit closer to the to the to the source And but if I separate the pinholes notice that the closer they are The more space the the the the the fringes are What we call the coherence has to do with the visibility of these fringes. So I have a maximum intensity and a minimum intensity. So This degree of coherence which we call the modulus of gamma is the intensity maximum minus the intensity minimum divided by the sum So for this source it is completely coherent. Why because i minimum is equal to what? Zero we're getting all the way to zero. So this goes away and this gives me one If i minimum is equal to i maximum, that is if we have a flat pattern then the coherence between those two points is zero Suppose now that I have two sources Let me bring this a bit closer And separate my pinholes a bit more So what i'm showing here is I have two sources at the moment. They're one on top of each other But now i'm going to separate them So I have the green one and the red one and it is not that they're different colors. It's just So you can distinguish them And they're mutually uncorrelated. So each one is going at its own at its own pace So one has its maxima and minima completely going from zero to a maximum quantity But as I shift the relative positions of one against the other The maxima and minima overlap Or they don't overlap and because When I integrate over time the cross terms cancel what I get is the sum of the intensity of each one of them So depending on the separation of the holes I can have Something that were the maximum of one cancels the minimum of the other and I have no coherence no correlation Well for smaller or bigger Separations I do have some correlation. So The correlation is not necessarily a monotonically decreasing function So I can have full correlation for two points here and I separate them correlation drops to zero and then it goes It it builds up again. It's actually an anti correlation and then it drops again and all that Turns out that if I measure the coherence as a function of separation like this That is a Fourier transform Of the shape of the source. So they have a connection with the shape of the source A continuous version of that Is let me skip this one. I'll go to the next one. It's the same So now suppose that I have a continuous source. So I bring this closer And it's not just one point I have an extended source each point here is emitting its own Light independently and creating its own fringes But then I add those fringes in intensity because they're They're they're incoherent with each other. They they don't have a temporal correlation I can see that as I as I open this There's a lot of visibility For small separations Then it drops drops to zero then I get a bit of correlation again But not completely the visibility here is about how much would you say Is this minus this divided by this plus this I don't know it's uh it's like about one One third or something like that And then if I keep making the separation bigger It it gains again and again and again and it also lays why because yes This the separation how can it be not monotonically decreasing? um It turns out that you can have uh So if each point in the source let me erase here So each point here Goes both ways and then interferes again here and create some Some pattern like this, but then I have another point that it does it again, etc There might be some combination of points here that for some separation give us a superposition of these intensity patterns Where it just happens that you fill the holes you you you the maxima and minima sort of disappear But as soon as you open your pinholes each one of these fringes Are gonna shrink they're gonna be more tightly they're gonna oscillate faster And that condition that guaranteed somehow that that you um That the maximum and minima cancel is no longer satisfied. Let me go back to the case of two sources. Maybe that's easiest to see there so Here it is So I have two sources Uh at a given separation so see initially, sorry Initially, uh when this is Let me bring this impression a bit closer When when the the two pinholes here and here are very very near the fringes Green and and and red from the two point sources are very similar. So their sum is very similar to each one of them As I make this different each each of those fringes is getting more more faster And at some point is so fast If if I made this a bit bigger, sorry So fast That as they shrink You hit the point where the minimum one matches the maximum of the other one And then I lose the coherence But if I keep going they get smaller again and they no longer coincide For a continuous source or something like that happens. It's just a bit harder to see But you're not guaranteed that uh the That once coherence goes to zero if you keep suffering your points your coherence is going to stay zero it it usually It oscillates so this uh Have you heard how who have you have heard of the michelson stellar interferometer? okay, so So let me show you this example and then I'll explain it with this so so if I have a source here of some size And then I fix my pink holes and I move the separation as I said the The correlation as I am I as a function of these pink holes it starts from one then it goes And it's completely gone then it goes negative And it's not so big then it's completely gone then it's positive and a bit smaller etc If I were to plot that correlation as a function of separation will give me Starts like this and then goes like this and like this and like this and like this What is this function? A sink is the Fourier transform of A rect which is the shape of my source Uh There are cases in nature where the source is so small that we cannot form an image of it But we can use the equivalent of a two pinhole experiment To calculate the correlation as a function of separation And uh And calculate this curve then we can do a Fourier transform and find the shape and size of the source And that's what michaelson did in mounts wilson in california and Forget the years With A telescope that had a big Like had mirrors like this So there was light that came here and then like this and then Same thing here And he he could make this Then a lens and form fringes here And he could control the separation of this it went all the way to about 10 meters And He could see how visible the fringes were here And with that he measured the Radial size of star core called beetle juice Which is so far and so small in the sky that you cannot Form an image to measure its size But by doing this interferometry and getting this as a function the coherence as a function of separation He could recover this curve And fitting it to what he knew had to be the correlation for for a disc because the star's gonna look like a disc What function would it be for a disc? What is the Fourier travel of a disc? Bessel one divided by its argument. This is our old friend the the jink the the airy pattern So by fitting it to an air pattern He could measure very precisely the radius of of the star So you can use coherence as an an indirect way of Measuring that There's an analogous thing in time with coherence time Sometimes you can measure with a michelson not stellar interferometer with the other michelson interferometer with a The one that divides thing in two paths the temporal coherence As a function of separation and if you Fourier transform that you get the spectrum So in some spectral regions, that's a better way to do spectroscopy Than trying to use Great things or anything else is more precise, especially in the infrared. I think It's what people use so you can measure coherence Easier than you can measure the spectrum directly. So gives you a nice indirect way of measuring the spectrum in the temples Case or this tiny little star in the in the spatial case Yes Yes Which one this one Well, uh, are you asking is does this really correspond to what I would get if I did everything? Or did I just use a formula? I just used a formula because otherwise it would be very slow and it would look exactly the same. Yes So I calculated it and then I put it in yes, uh, but you could do it And and you can see that it's a Fourier transform because it's a superposition of different waves Uh, then it gives you something like a Fourier transform Yes, no, these are just simple light waves solutions of Maxwell's equations They're analogs for showing your equation, but this this is just good old Maxwell. It's a classical waves Decoherence in the quantum sense Yes, uh, I'll have to think about that because yeah, this is this is a fully classical picture So here coherence is relates to uh Just temporal correlation of random events giving rise to this field But yeah, you read the same mathematics apply to to quantum decoherence and things like that. So Does that concept make sense? So, uh coherence then it Um has to do with the ability to uh Of waves to interfere with each other or not And let me stress again. They always interfere with each other. It's just whether that interference is persistent in time self consistent in time enough that when you do a time average You see a significant signal or you see an average that is is now gone Okay, so how does this relate to? uh microscopy so I'm going to draw the same drawing that I've drawn every time Have some sample here then typical system Have some sort of lens or lens system does a Fourier transform takes me to a pupil then another lens and and here So there are two types. So this is our detector We always talk of these two types of of uh imaging and they are the extremes Some context we use the intermediate one But for microscopy you we usually talk about coherent imaging or incoherent imaging When do we have coherent imaging? So yeah, so we have some sample that is uh Transparent or semi-transparent something and we illuminate it with a laser source. So each point here is going to Defrag the light in some way and all that and we form an image, but it's so organized That their effects here are persistent in time On the other hand if we illuminate this with white light or with a An LD or something or even if we illuminated that with a laser But what we're observing is not the laser itself But say the fluorescence which is a more random emission Then that's incoherent that that means that each point is it's like a dock Each one is going to meet its own its own wave That over time over the time of integration of the detector is not going to interfere with uh with that of the others And and how we model those systems is is very different If this were coherent How will we model the the the image? Suppose that there are no aberrations. Let's start with the case of no aberrations So to go from here to here We roughly what what type of mathematical description a Fourier transform because this This is the focal distance here. So we roughly do a Fourier transform So if I have a Uh, I have a Fourier transform here Then here I have To go through the pupil. Let me call that multiplication by some function p And then from here to here Another Fourier transform or an inverse Fourier transform if I invert by axis so that it looks like an inverse Fourier transform So I have an inverse Fourier transform And this is applied To the initial distribution And what I measure so this coherent What I measure is what the intensity which is equal to Mod square of this So intensity would be this So this is coherent Source if it is incoherent Then it's different because Yeah, so so then I would have to look at each point here Each point here, uh Um Would be an independent source. So I would have let's say a delta of x minus x naught That's the field being emitted from here and to go to here. I need to do a Fourier transform of this and with the Fourier transform of a delta It's a constant and if I shift it is a linear phase Then I multiply this by the pupil Then I um I inverse Fourier transform this Then I mod square this and then I have to integrate this times my Uh Let's say of x zero My initial intensity Squared Integrate over all the x zero. That is I would have to to Add the intensities Of each contribution for each point source and if you do the math this turns out to be just the convolution Sorry, this is not a vector here the convolution of the intensity of the initial field with something that is the Auto convolution or more precisely the autocorrelation of of the pupil something that is Uh the Fourier transform actually the Fourier transform of the pupil Squared So this is the Fourier transfer of the pupil would be my every disc If I mod square it and then each point Is causing one of those shifted at a different location And and and it gives me that Because this is a convolution. I can write it as the inverse Fourier transform of what type of operation a product product of the Fourier transform of the first thing of the intensity times the Fourier the the the Inverse Fourier transform or the Fourier transform of The square of the Fourier transform of the pupil That's called the autocorrelation and we give it a name which is called the OTF The OTF has a nice interpretation graphically suppose that these are pupil OTF is a function of a pupil coordinates And what it is is if I have these people here I at point row I come to that point row here And I draw another copy of the pupil there and it gives me This overlap area So that function This OTF is maximum when row is equal to what? zero and it drops to Zero once I have the separation is twice as twice as much as the radius of the pupil so just to Stress the difference between coherent coherent in coherent imaging you have to Fourier transform the initial fields Multiply it by the pupil inverse Fourier transform mod square here. You have to Fourier transform the initial intensity Multiply by this OTF an inverse Fourier transform And it turns out but because this is this drops slower than the than the pupil There you can transmit more frequencies more spatial frequencies in a system when you're using coherently illumination than coherent Why because I have to this row is twice the radius And you still have contributions while we've coherent imaging you only carry frequencies up to the radius Of course it drops So this this OTF for a perfect system if I plot it looks something like Looks pretty much like a triangle and then it drops down like this There's a close from formula Of course if we have aberrations We have to include those aberrations in our pupil and this gets horrible So let me illustrate this with this Okay, not this one So here, uh, I'm gonna make this bigger. Okay, so now you all know how we create these movies So here what I'm doing is I'm taking our friend Joe as my object And this is coherent imaging. So what I did to model this is I took Joe Took it for your transform Multiply it by this aperture inverse Fourier transform Mod square that's coherent imaging. So for a big For a big pupil like this all of Joe's Fourier transform fits through the pupil and I get a sharp image Of course, if I make my pupils smaller I'm starting to lose lose the high frequencies and the image gets worse and worse And it gets bad in a very wavy way. We see a lot of interference here We see this this this fringes here because things go in phase out of phase and we have these interference effects On the other hand if I do incoherent imaging what I need to do is Um Take the intensity not the amplitude of the object Fourier transform that So here I took Joe not Joe, but Joe squared and I Fourier transform that Then I multiply that by By this m t f or this o t f that looks like this is this overlap between the pupils When this is at the center is here This is largest is the center there where the cursor is As we displace this in any direction it drops to zero nicely And then I take that Fourier transform of the mod square of Joe multiply by this Inverse Fourier transform and that gives me the the the result here. Let me make this one bigger again And it's similar that if I make this bigger the image is better but The smaller it is The image starts to get bad But it's not wavy. It's just it's just blurry But it's blurry in a more in a less is it has less artifacts in a way so So while it's not good Uh It's not introducing features that you don't know if they're interference features Like something's here or if they're real features. It's just a more Blurry image Yes Of sufficiently large pupils. No everything is getting through. Yeah Yeah well, I am This is a an ideal world's case In practice your microscope your lens is going to have little particles of dust or something and those are going to introduce some coherence Interference effects that you're going to see as fringes And those are going to be very persistent if your light is coherent because they're they're going to Be very visible if you have an incoherent source Those are going to be spread out through the image and their effect is going to be sort of blur away So the the real life effects of the fraction from the From from other parts of the system or from from particles, etc That that is better in an incoherent system than in a coherent one But other than that in an ideal world. Yeah, if if if all the Fourier transform of the object fits through the pupil Oh strictly speaking the coherent is better because Everything's fitting through a flat pupil. So you're multiplying it by one essentially and it gets through in the coherent case You're multiplying it by something that Drops so it's really doing something You cannot have a big enough pupil for for an incoherent system because it's always the empty of his Dropping a little bit. It's best at the center and it drops a little bit But it's it's still pretty good So you lose some but you win some in that you average out any persistent coherent effects from defects in your system Is that sort of what you were? So I wanted to um Just finished by mentioning something else so Again, so for coherent We Fourier transform pupil inverse Fourier transform mod square If I had aberrations, where should I throw them in? Sorry Here so I would put here aberrations And incoherent I have mod square Fourier uh OTF Inverse Fourier Where would I throw the aberrations? They're sort of hidden here in the definition of OTF because when I have this this formula I will have to put them there Now the problem with modeling aberrations in this system is uh Something that we were talking about at the end of class uh with some of you I can only model this with aberrations for certain types of aberrations. It really has to do with your question as well What type of aberrations? Still let me model my system this way Remember that we had aberrations that depending on row or row and h or et cetera So the aberrations that depend on h Mean that the system is going to behave differently for points here than for points here than for points here That breaks the shifting variance of your system And once there's no shifting variance This magic formula Fourier transform Transfer function inverse Fourier transform This linear shift invariant system thing no longer works So the only aberrations that you can still model with This type of thing are the focus and spherical aberration Because they don't depend on On fields they don't depend on on Where you are here this is they they do the same damage here than here than here than here Because otherwise Once you do the Fourier transform You don't have the dependence here. You Fourier transform that there's no way you can put the aberration here that depends on this point so Unfortunately, uh, you can only model Simply aberrations if it is spherical aberration and the focus that's it. I mean the others Uh, you have to work harder to to model because they break this shift invariance um so, uh, Anna concertini was mentioning this Uh, how did she call it isoplanetic approximation in that The aberrations only depended on on the pupil coordinates not on here That is the approximation that lets you model things easily Once the approximate the aberrations depend on where you are here Then then it's it's it's it's much harder okay, um So I guess that's What I have, uh, do you guys have any questions? We're happy to answer any questions or try to answer any questions If not, uh, you know where to find me and uh, And I will have a slider a slightly longer coffee break and uh, oh, thank you very much