 So now, Vaso Miguel Alonso, we start with aberrations. You mentioned a lot of these aberrations evaluated by means of rays. Yeah, rays and waves, yeah. And also I'll do a bit of waves. Ah, waves. Okay. All right, very good. Good morning. So before I start with aberrations, I wanted to finish quickly what I was discussing yesterday. And this morning I was writing some code to show you something, so let's discuss that. So we're talking about the point spread function. Well, I didn't say the word point spread function, but it's the point spread function you can think of as when you have the beam focus like this, there's an intensity maximum here. If you look at the slice, this gives you some function like this. So this is the intensity that corresponds to a point source somewhere in the system. So if you have the system, I write this diagrammatic system here with a pupil. I have an object here, and then I have the image here. This point spread function gives me something like this. And we calculated that this is proportional to, on this plane, is J1, the vessel of K. Let me write it as 2 pi divided by lambda rho, which is the radial variable, times DNA divided by the same thing. 2 pi over lambda rho times DNA and all mod squared. So this gives me the width of this distribution. And because the first zero here happens when this argument is at the value of 0.609, then we can measure the, and that's the distance at which you could have two of these things superpose, such that you can tell that there are two distributions. So I'm going to define half of that distance as the width of the point spread function, because then I could put another one right next to it, and they wouldn't overlap, and I can still distinguish it. So this is sort of a definition of the width of this point spread function. So that goes, then, rho is about lambda divided by DNA. And then there's this 0.609, and there's a 2 pi somewhere. So this is the width. Now, what we didn't say is how long this thing is. So I want to start calculating that. And why is this? Because of several aspects. One is what we mentioned before is the depth of focus. If I'm going to form an image, how much can I defocus and still see a fairly sharp image? That could be good for some applications. For other applications, it's bad because if you want to image several layers, you just want to focus on a given layer. So what is this distribution here? So for that, we remember that if we think of the field as caused by a bunch of plane waves, I can always decompose my field into a bunch of plane waves like this. And let's ignore aberrations for now. We'll talk about aberrations later. And they all interfere here. What is the field distribution on this plane? So by the way, what is the maximum angle here if I look at this angle here for the last wave? Do you remember what the name of the sign of that angle is? Another name is the numerical aperture. So what I'm going to have then is an integral of plane waves e to the i, k, this unit vector dot r. And then I have, if I go to polar coordinates, I have u, du, d phi. But because this is independent of phi, well, not yet. So I have an integral from here to this u to the na, and this goes from 0 to 2 pi. And we calculated this formula by taking this exponential and evaluating at z equals 0. Now we're going to do the opposite. Now I'm going to say, what if x equals y equals 0? So I'm going to calculate this on axis. What happens to this exponential? So I have 0 to 2 pi. I have 0 to na. So what is this? This is ux times x plus uy times y plus uc times z. How many of those survive? Just e to the i, k, uz times z. Then u, du. And what happens to the integral in phi? Nothing depends on that angle. So I do this, and I replace this with 2 pi. So this is u at 0, 0, c. And remember, uz related to this u is 1 over u squared, because this is the radial thing. So I'm going to change variables. If I say uz squared, let me do it this way, uz squared is equal to what? U1 minus u squared. So if I take the derivative of this, I have 2 uz duz is equal to the derivative of 1. That's 0 minus 2 u du. I can cancel it, too. So I can turn this into an integral of e to the i, k, uz, z. And this u du is the same as uz duz. And that's a pretty simple integral. It goes from where to where. U is 0, uz is 1. So that would be down here, but I have another minus sign that I haven't used. So I'm going to use that to flip this. And then when u is na, this is just the square root of 1 minus na. Can we evaluate this integral? Yes. If I didn't have this here, it would be really easy. Because it's just the integral of an exponential, the exponential divided by that. We can use a trick from Fourier transform or the equivalent which is saying, oh, this uc I can think of getting it by taking the derivative with respect to c of this exponential. So I can replace this by a derivative with respect to c divided by i, k. I pull that out. And then that gives me the answer. The other way to do it is to ask Mathematica. The lazy way to do it. So I'm going to... Yesterday when I asked Mathematica, the other integral, it gave me a crazy answer, but this one I've tried it before, and I know that it gives me an answer. It's not too bad. It looks like that. And I use the fact that na is sine of this theta. So this cosine theta is that. So you can calculate that. But what I want to show you here is that that is that plot if I take the mod square of that. And I'm showing two curves. The other one is the approximation if I remove this uc, which is just a sinc. So this u0, 0, z is very close to just being the sinc of sin pi z divided by lambda times 1 minus cosine theta. All divided by the same thing, pi z lambda 1 minus cosine. So I'm trying this for different thetas and look at the two curves. They look a lot like each other, only in the limit when theta is very, very large. You can see that the blue curve and the yellow curve are different. But they're about as equally wide. So if all we want to do is estimate this width, then that's fine. I can use that approximation. So how wide is this? What is the distance to the first zero? What value of c does this go to zero? So when z is zero, no, because this divided by this gives me 1. So I need all this stuff here multiplying pi to be equal to 1. So that gives me lambda divided by 1 minus cosine theta. Or in other words, lambda divided by 1 minus the square root of 1 minus na squared. Assuming that we're in there. And that lets me calculate this distance here. So that's the distance. In fact, just as before, I'm going to use half of that distance to put a say where this fits. And I can draw a little box here. So if I have a beam focused with this angle, I can now draw a box where I know that the focus, the focal region is. And the length here, or the length here is going to be precisely lambda divided by 1 minus cosine theta. While the width here is lambda divided by, well, let me show you how that works. So here I defined the integral. So I can calculate this, for example, for pi where four is going to calculate numerically because there's no closed form expression. The intensity of the field takes about 10 seconds. There it is. So this is for an na, well, for an angle of 45 degrees. This is the cross section of the intensity pattern. And this is the box where the focus is. The longitudinal direction and the transverse direction. And it's always that half the distance to the zero, half the distance to the zero. We can see that for an na like this, one over the square root of two, we have a box that is much longer or significantly longer than wider. The units here are wavelengths. So the width is about a wavelength. The length is about four wavelengths. What would happen if I make this, say, pi over eight? Will the box get thinner and gets longer and it gets also wider? Because this width here, I have it here. This width here is 0.609 divided by sine of theta. So the transverse width scales as the inverse of sine of theta. The longitudinal width scales as the, oh, here it is. Sorry. Sine theta and then 0.609. So the transverse width scales as the inverse of sine of theta. The longitudinal width scales as the inverse of one minus cosine of theta. For small theta's, this is linear in theta. For small theta, this is, for small theta, what is cosine? One plus minus because it has to be smaller than one. Theta squared divided by two. So one minus cosine theta is about theta squared divided by two. So it's quadratic. It grows quadratically with theta. Well, the other one is linear. So let's see. So as I close the numerical aperture, the focal region does get wider. It looks thinner, but it's because I'm using a bigger scale. I'm scaling things so that it fits. So it gets wider, but it gets much longer. And on the other extreme, if I do pi over two, what numerical aperture is that? That's one in there. That's pretty difficult to do. Pretty much impossible to do. So that means you have all plane waves coming even in this direction. But mathematically, it's very easy to do. I just put the numbers and let it run. So that is the extreme case of a very, very high in a field, and it gives me a much square thing. It's never square, but for very high in a, the longitudinal length of the focus, the longitudinal point spread function is not too different from the transverse one. Not fairly clear? So if you want to do something that is imaging in 3D, you really need high angles. You really need fast lenses, a sharp focus. That gives you much more longitudinal localization. So what we've talked about so far is in the ideal world where everything works beautifully. So what I'm going to discuss now is this concept of aberrations. Online, I posted some notes that I prepared for another course where I had more time to do this, but I figured that these notes could be of interest to you. I give some explanation of ray optics in general, this concept of phase space, geometrical optics in the paraxial limit, how to trace rays, the matrices, etc. So I'm going to skip all that now, stop some pupils, and I'm going to come to the aberrations. So theory, and I'm going to give a simplified description. This is a more rigorous description, I'm going to give a simplified description, I'm going to stop here. So let me draw the figure that I'm going to use. So again, suppose that you have a plane with your object. This is the axle propagation, I'm going to have an image here. And a cartoon version of any microscope or optical system, then as we said yesterday is a lens that does a Fourier transform where we have typically a pupil, and then another lens that forms an image. So this is the canonical 4F system where this distance is F, and these two Fs don't need to be the same as these two, but that's okay, just to give us an idea. So we have some object here, this does a Fourier transform onto some pupil plane, there is where we limit how much light goes through the system because of reasons like aberrations or reasons because the lenses cannot be infinite, and then we form, we undo that Fourier transform with another lens. In the ideal world, if I have a point here that is sending light, all this light would come out parallel. And then some is blocked, some keeps going, and then this is focused very nicely here. And there's one that gets here. The only effect is that we're blocking some light with this top here. However, in the real world, I'm going to use pink for the real world. Well, before I go to the real world, the ideal world, what is the shape of a wavefront here? If the rays are going like this. Sorry? Yeah, it's a plane wave. Wavefront is perpendicular to the rays, so perfect wavefront there. And if I move this to the side, it would be perfect, it would just be tilted. In the real world, it is not perfect. Why? Because the lens is not perfect. Because there's a lot of other problems. So the wavefront is going to be some funny shape here. And that's without counting the errors from this or the lens. Now, how do we count those? Well, you could, for the error that we're going to do, we could trace back this and see what those errors are and also accumulate them here. So when I draw this tilted wavefront, it's not only accounting for the errors of this lens, but it's like for the errors of both lenses, imagine that I propagate all the way to here, and then in my mind, I propagate it back to the perfect lens, and then I see what the wavefront would be. So that would have the errors of both lenses. So I have some deviation here. And this deviation I'm going to call the aberration, W. And it's a function of two things. So if the coordinate here, typically people use for the pupil a coordinate rho, which has a component x and y. So rho is rho sub x and rho sub y. These are the pupil coordinates. And normally they're normalized so that they go to one at the edges. But that's not very important. So this W is certainly going to depend on... I want to use my same notation there. It's going to depend on rho. But it's also going to depend on something else, which is... I get this aberration when I'm looking at this point. But if I'm looking now at this other point, so this one would ideally go like this, but again, this rays might do something funny. And I might have then some funny shape like this with respect to the ideal, which is like that. So for this point here, the ideal wavelength should be like this, but the difference from the ideal might be like this, and it's a different shape. So I call this also an aberration. And this difference is not necessarily the same as the difference here. So the difference depends not only of where in the pupil you are, but also of what object you're talking about. And this point I'm going to call h. It's called the field point. It's the coordinate of a point in the object that I have. So h, which is h of x, comma h of y, which we can also call just x and y, is called the field point. It's called field point, but it's really a point in the object or a point in the image ideally. Yes. The deformation is being careful. So I would turn it around and say, I have no good reason to say that it would be the same. So because the light is going through different parts of the lens and it's going at a different angle through the same lens, this is traversing more glass than this one. So different things are happening to this race and from this race. And that might cause a different type of deformation when you're going through this lens in this direction than when you're going in this direction. So I just cannot be sure that the deviation is the same. So just to be super careful, I'm going to say, yeah, I'm going to say that this could depend also on this field point. It might be that it doesn't. It usually does. So then we have in general this wave aberration function, w, that depends on h and depends on rule. So in the ideal world, w is if you could go to the lens store and tell them give me a lens with w something, what would you tell them? Zero. Yeah, give me a lens with w zero. And the guy would laugh in your face. But in an ideal world, the aberration would be zero. The aberration is the distance between the real wave front and the ideal wave front. It's a deviation from ideal behavior. So is it clear what this is? The problem is that dealing with this in general is hard because we don't know what that function is. So whenever we have a function like that and we're optimistic, what do we do? What can we do? We can see what is the simplest way to model that behavior. If you have a complicated function, you want to model it within some region in a fairly straightforward way, you can do an expansion, a Taylor expansion. So that's what we're going to do. We're going to perform a Taylor expansion of this, assuming that there's a part that is independent of h and rho, a part that grows linearly, quadratic, et cetera. So just do an approximation. So the first part is w. I'm going to call it 0, 0, 0. And you'll see where this name comes from. So this is the part that is constant. And this one we don't care about. We can always choose it to be zero. Because it's just a constant shift of the wave front. We don't care about it. I can do a variation that is linear in h or linear in rho. However, there's a problem with this. If I were to do a Taylor expansion of quantity like this, that is a scalar in terms of vectors, the linear term, there would be some number here times, let's say h. But this doesn't work because this is a vector. The only way to make this a scalar would be if this itself were a vector and I had a dot product. But then this deviation would have a preferred direction. I'm going to assume that our system is rotationally symmetric. If the system were not rotationally symmetric, I could have this term. But if it is rotationally symmetric, it cannot have a preferred rotation. So things that are vectors are forbidden. That doesn't exist. I don't have anything that just depends linearly on h and not on rho. That's not allowed. What I can have and same with rho. So the next term, I'm copying what I'm doing there, is what happens if... Now I'm going to look at the quadratic terms. So I have one that goes like rho. And because I can have another rho, that's good because I can have rho dot rho. And this is a scalar. This is good. This I can write as rho squared. I also have a term that goes like rho dot h and a term that goes like h dot h. And I'm going to use a notation where the subindex, every time I have an h square like here, I'm going to say I have one of those. Every time I have a rho square, I'm going to say... What is the convention here? I see. I have two of those. Sorry. So this is... I have two of these. I have two h's. Then, so what matters is, this one says I have two h's and no rho's. And this one, I have two rho's and no h's. And this one, I have one and one, but they're in the form of a dot product between them. So I have one dot product between them. That's what this calls me. So here I have no dot products between them. Here I have no dot products between them. So that's the convention we're saying. So then we have w, n, m, l, let's say. So powers of rho of h, powers of rho, and then powers of rho dot h. So it's just a notation that we're using. So we had the constant term we could have, but we choose to be zero. Linear terms, there are no linear terms because of the symmetry of the system. I can raise these things. Then the quadratic. And I did all the combinations. Rho with rho, rho with h, h with h. Then I keep going. These are the simplest terms that will have an interpretation. We'll talk about their interpretation. Then I can keep going with the cubic terms. How many cubic terms can I have? I can't have a bunch, but I have none. Why? Because of the same reason. Because if I had, say, two rows and an h, to make this a scalar, I can do the dot product between two of these. But then I'm left with a vector on its own. And this needs to be dotted with something. So there needs to be a special direction. And if my system has rotational symmetry, that is not allowed. So I skip, in fact, all the odd orders here. I just jumped those. So I go directly to the fourth order. So I start with the one with rho dot rho squared. Then I have one with rho dot rho rho dot h. So all four things are rows. Three things are rows. One is an h. Then I have one where I have two rows and two h's. How many ways can I do two rows and two h's? I can have rho dot what? Rho dot rho and h dot h. Or I could do dot h times rho dot h. This is the same as where in this. Then, so I have all four rows. Three rows and one h. Two rows, two h's, two rows, two h's. Then what am I missing? Yeah, so rho dot h times h dot h plus all everything is h. So h dot h squared. And now, so we don't cheat. Move this away. Let's write the subindices. So what are the subindices here? Please help me out using those rules. 0, 4, 0. Someone else? This one? 1, 3, 1. Because I have a dot product. This one? 2, 2, 0. Because there's no dot product between the two. This one? 2, 2. This one? 1, 1. And this one? 0. How many dot products between rho and h? 0. Okay. And we could keep going, but most lens designers say, okay, this is enough. I'll stop there. These are the ones that I want to look at. Yes? Yes? Yeah. Yeah. Ignoring. No, no. Comma is here. There's a usual confusion in here between, and this will, I hope, will be clarified soon. Some people say first order aberrations and third order aberrations. Some people say second order and fourth order. When they are the same thing. It depends where you're referring to the wavefront error or the transverse error. Turns out the transverse error we're going to see later, or actually very soon, is the derivative of this with respect to rho. So it's going to bring down the order. So one of these guys is Comma. Okay. So, yes. Then good luck with filling all the, yes. Then you would have a bunch of things. Because you would have many more even at this level. Because you will have rho dot a vector times rho dot another vector. So there's a lot, lot, lot more combinations. And normally we say that's not a problem because it's so much easier to do lenses that are rotationally symmetric. It's difficult to make cylindrical lenses. And why would you? Well, that's no longer true. There's a whole field called freeform optics where you can now, because there's very precise manufacturing and polishing tools, build lenses that are very, have arbitrary shapes. And you have a lot of more degrees of freedom. You can make your system slider more compact. And because oftentimes they have mirrors and you have to fold the system like this, you're working off axis. That means you have to make your elements not rotationally symmetric. And then this theory is not very useful. But thankfully microscopes are not there yet. So most microscopes do use rotationally symmetric lenses. So for the purpose of this course, this will do. And even if you were to learn that, it's always good to learn first this case and then to appreciate how beautiful and simple this is. If you think this is ugly, this is nothing compared to the other case. So is it clear that we just did what we had to do now? Now let me come back to the drawing. I'm going to erase some of these rays. And I'm just going to look at one. Suppose that for this one I have my ideal wave front, touches this, and then I have my aberrated wave front. So the ideal wave front, and suppose that this angle is not very big. Let me make it a bit smaller. So for the ideal wave front at a point here, the ray goes perpendicular to the wave front in this direction and this is going to end up somewhere here. And where it hits here depends on the direction or the position on this side before the lens. If I move to another ray here, is it going to hit a different point or the same point? The same. So to see where you hit here, you only need to look at the direction here. So if the aberration causes an error in where you're hitting, it is not because this is shifted in this direction, but because it is tilted. If the wave front is tilted, then that causes that the ray goes in a slightly different direction and then ends at a different point. This transverse error we call epsilon is a vector and that's the transverse error. And what is the relationship between this error to the wave front aberration? It's related to the angle. How do we calculate the angle? If I have this function as a function of rho, what operation do I need to know to do a good approximation to the angle? And that goes back to your question. The derivative. If I take the derivative, how much this is changing as I go up, that tells me what the difference in angle is between this and this. So that's going to give me the deviation of direction here that turns into a deviation in position here. So I'm not going to be very careful with the factors. They're all in the notes. I just want to transmit the idea. The errors here as a function h and rho are proportional to the derivative of w with respect to this coordinate, the pupil coordinates. And it's a vector derivative, so I'm going to write it like this. So the transverse errors depend on this. So let's come and look at these different terms. So what do I get when I take a derivative of this one? With respect to rho, I get, so I just get a 2w020 and rho. It's proportional to rho. What about the next one? So it's w111 and h. What about the third one? Zero. So this error that is independent of rho, this only depends on something called piston, that doesn't do anything to the image. So let's forget about that. And the same is going to happen with this one, of course. Plus let's go with the next row. So what do we get here? 040. So I pick up a 4 and then I have rho dot rho times rho. Plus w131. And here I have a more interesting expression. What happens when I take a derivative? Well, I have rows in two parts. So I can take the respect to this part or respect to this part. So one gives me rho dot h times rho. The other one gives me what? Two times... No, this is the one with a 2, sorry. Plus rho dot rho times h. So I've done that one. W220. What do I get now? Derivative for this one. No, because it still has a rho dot rho. So I pick up a 2 and I have h dot h rho. This one here, w222. I have 2 and then I use the chain rule. So this is rho dot h times what? H. And I just have enough room. I'm missing 2, but thankfully this one is not counting. So what is the last one? 311 h dot h h. So aberration theory is, in a way, all about doing an extra expansion. And then giving names to each term and getting to know them well. The Cernike notation? Yeah, no, this is the Seidel notation. Cernikes are when you use polynomials and that is more... For those, it is difficult to insert the dependence on h. So when you have something that is very collimated, for example, then it's good to use the Seidel because you just do an expansion in this or some kind of polynomials into that. But if you want to look at the dependence on position, transverse position of the object, something like Seidel works. There are people that have done Cernike versions of that, but it's not very common. Anna? German. Cernike. Fritz Cernike. Yes. They are mostly in the there. And by the way, when you move in the form of the after the after the after the after the after, it's of isoplanatic, we call it isoplanatic signal. This is an ideal signal. Yes. This is the question. Isoplanatic is the more inextricable than the rule, that all systems are not isoplanatic. Yeah. So yes. So that is the approximation which things don't read. You ignore the dependence in H. This word isoplanatic and the word another relation between isoplanatic and aplanatic is. Yeah. Aplanatic is. Okay. Because there's a term aplanatic that I'm going to talk about later, which is when things don't depend on the main aberrations that are linear on H. And there's a funny thing that I learned from some Spanish friends that are lens designers that are also very interested in language. I always thought that the word aplanatic or isoplanatic came from, maybe isoplanet is different, but aplanatic I always thought that was, that is not on a plane. But they say that the word they use for aplanatic in Spanish is not aplanatic, which is what we would think, but aplanetic. Because they say planetary comes from planetary. Planet means something that moves, something that migrates. And aplanatic is something that doesn't move as you move this thing. So it does not come from plane, it comes from planet, which means traveler, something that moves. So just a bit of trivia. Okay. So let's then look at each one of these terms and what they mean. So the first one, and we're going to look at both the wave aberration form. So this is what we call the fourth order theory, so second order and fourth order. But sometimes we call it first here when we're talking about transverse. These are linear in grown age and third when we're talking about the aberrations. So what are the meaning of each one? So W020 goes as rho squared. That tells you that your wave front here is independent of H. And all it does is it increases quadratically. What does that mean? So if I were to draw a quadratic here, how would it look? More or less spherical, it would be something like this. That means that your wave front is not flat, it's a bit curved. And that means that your image is not going to form where you expect it to form, it's going to form closer or further away. So you did not focus your system well. So we call it the focus because the wave front is a bit curved. So this is called the focus and it causes the wave front to be a bit more curved or a bit less curved as you're approaching the intended point here. So rather than having this curvature, it has a tighter or a more loose curvature and then you're not focused where you should. So this one is called W020. It's called the focus. What about W111? So this tells you that let's look at it from the point of view of the transfer separation. So ideally this would take you to the image point, but this is going to essentially tell you your variation with, where is it? Here, your wave front aberration is linear in row. That means that your wave front is, how would respect to the ideal wave front? What is a linear equation? It's just tilted. So it's just, you didn't get the direction right. It's still flat, but it's tilted. And that tilt is proportional to H. That means the further away you go here, the more tilted it is. Therefore, your image point is going to be somewhere else and if you go further away it's going to be somewhere else. This is an error in magnification. When you calculate the magnification of your system, you got it wrong. This is a small error in magnification. So W111 is called tilt. And it's a magnification error. Is that fairly clear? So those are the easy ones. And usually we don't even call those aberrations because if you redefine where your image plane is and what your magnification is, then you make them zero. Just that you got it wrong and well, you can get it right now. The other ones are more fundamental. So 040, this one here is called spherical. Spherical aberration. Why is that? So let me draw this again. So I have my ideal wave front. And I have the ideal rays that would go here and would converge to this point. So down here the wave front would be a nice spherical wave front. However, my deviation here is a quartic. It goes like row to the fourth. A quadratic we know is something nice and parabolic or more or less round. Quartic is much flatter here but then it takes up. It goes like this. That is, my wave front here is going to be, if I come a bit further away, rather than being this, over here it agrees but at the edges it curves a little bit more or a little bit less. That means that the central rays do focus nicely where they should but then as you go off axis of the central ray, they start missing the point. If you go this direction, if it goes this direction, then they open in the other direction. So this is a fundamental aberration. That means that the previous two told you, yeah, all your rays are going to intercept but you got it wrong laterally or longitudinally. This is, no, your rays are not going to cross. They're not going to cross all at the same point. So this is, the spherical aberration is the more serious one because of what? And this goes back to your question. This one is the only one that exists even if you're at the center. If you're at the center because it's independent of H, it still exists. All the others disappear. All the others start giving you a problem as you go to the sides. But this one is a problem throughout. And that's not good. So let me erase here. Erase more than I wanted. So 0, 4, 0, spherical aberration. And this one tells you that your rays are going to cross like this but then they start crossing. The more outside you go, the closer they focus or the other way around. So you have them cross like this and then this one starts crossing further away, further away. So it causes the rays not to cross. And it always places it so that the focus of the central rays is at the intended plane. So you can alleviate the effects of this aberration. We'll see later by putting a bit of the focus to move you from here to what is called the circle of minimal confusion. So where the intersection of all the rays is the smallest. I'll show that in Mathematica. So this is, again, the most important one because it happens even when you're at the center. The next one that is more important is this one, W131, and this is comma. And there's a nice analogy we can make here. So for example with spherical, it looks more like comparing this to the second order. So the focus is rho squared. So spherical is at the focus that depends on where you are. So the more outside you are, the more the focus you are, and that's why the rays are focusing more for the external rays than here. Well, comma looks like at the focus that depends on the field or looks like a tilt that depends on where you are in the pupil. And this one does a strange thing. At the center is zero because H is zero. But as you go away from the center, if you look at the lens, the rays going through the center of the lens, they hit the intended point. But the ones that move in a little circle around it, they hit a circle that is also shifted. And then a more external ring of rays describes a bigger circle that is even more shifted. And it describes this very characteristic shape that looks like an ice cream cone or something. And if it is positive or negative, these things will point outwards or inwards. So at the center it doesn't have an effect, but it increased linearly with distance from the center. And linearly is quite fast. So this is the second most important aberration. So the aplanadic system that is corrected for these things does not have these two. So it suppresses the two more important aberrations at the center of the field. So this one is comma. And it causes the rays, instead of hitting a point, we get this characteristic shape like this. And maybe it's better if I, at the same time, show you this. So I'm going to show you on one screen the mathematical notebook and the other one there. Ah, it is here. Let me close this. Okay, so here I define my wave aberration function. And I define things with rays. And let me show you this. So this is a diagram of different rays coming from, say, the second lens and hitting the intended plane. Here I have a lot of sliders. So z, let me move my image plane back and forth a little bit. X, 0, and Y, 0 are essentially h of x and h of y. So if I move those, I'm moving my ray find in one direction or the other. Na is the na. I'm going to leave that alone. So what does the focus do? W is 0 to 0. It makes my rays focus before or after. So the focus is still a perfect focus. I just need them to redefine what my image plane is, and I would correct that. Here W111, which is a magnification error, is not doing anything now because I'm at the center. So I have to move a bit to the side. So I see this position here. And if I move W111, I see that it displaces a little bit my intended image. It's still a perfect image, but it moves it a little bit. So that's telling me, yeah, you got your magnification wrong. Now the aberrations are the more interesting ones. So for example, if I put spherical aberration, I can come and look at it here. You can see that the central rays are focusing at a point. But the more high angle rays are focusing faster or slower if I go the other direction. So my image is quite bad if I look at it from here. But I can correct for it a little bit if I move my image plane backwards to this minimum confusion disk. Because that's where the final rays that I have have the smallest cross-section. Coma, okay, let me do this. I'm going to run this again so it resets my values. What happens now if I add Coma? What will happen? Any guesses? Nothing because I'm at the center. So if I add Coma, I can add all the Coma I want. Nothing happens because I am thinking of an object that is at the center here. I have to move off the center to see some effects. So let me move, let's say Y here. I go up and now I add Coma. And then I get this funny shape here that is hard to appreciate. I will show it to you from another angle in the next graph. So this is showing the same thing, but just in terms of the cross-section. So again, I can move sideways or up. And then if I do focus, I just make my rays not hit. But I can shift, I can correct for that by just moving my image plane a little bit. However, if I add spherical, now I can see that even if I do focus, I don't fully correct that because some circles of rays are moving in while others are moving out. The best I can do is choose this. After that, it starts getting big again. So there's a minimum confusion. And for that reason, some people like adding some amount of the focus to spherical so that you can automatically get the minimum error. And of course, the effect of this, I should have said, is that your image is going to be bad because instead of having all the rays from a point going to a point, you're getting it going to a big area. So it's blurring your image. So what about spherical, coma? So I add coma, nothing happens because I'm at the center. But if I move off the center, I get my little light-screen cone there. This direction or this direction. Or if I go up or down. And this is for positive coma. For negative coma, what will happen? The at-screen cone goes the other way. So it points towards the origin or away from the origin. Coma. So the next one is field curvature. So this is... I'm going to put this here. Oh, I changed the wrong one, sorry. This looks like the focus that depends on where you are. So if you're at the center, this goes quadratically. So this takes a while to pick up. But it tells you that if you're at the center, you're fine. If you move your point a bit away from the center, you get some of the focus. If you move further away, you have more of the focus. So that means that's for... Let me erase this. Where's my eraser? For this point here, everything's fine and I have my nice image here. But if I come... So this goes nicely here. But if I come here, I'm going to have a bit of the focus that depends quadratically on this distance. So over here, I'm going to be the focus. That means my rays are going to cross here. If I came to this side, where would they cross? I would like them to cross here, but they won't cross there. I'm also going to cross here because the focus depends on my distance from here. If I move further away, the focus is bigger and same here for this point here. So what does this mean? That I do have a perfect image. It's just not on a plane. Because all these points, if I join them, I have here. So my field where I form a perfect image is not flat. It's curved. And it's curved one way or the other way, depending on the sign. Nowadays, we couldn't theory correct for this by making a CCD that is curved. So I mean film is flat and there's not much you can do about that. But you could make for a CCD or something that is curved and they would compensate for this. So let's look at, so I should write here. So this field curvature. So we start with this. We have our imaging system and this images onto a surface like this rather than to a flat surface. You have a question? So let's try that here. So if I add 2 to 0. Look, so now I move my rate. So this one is fine. Ah, it looks very nice. I wish you could see that. So at the center is fine. So I added some field curvature. And as I move up, this focus is before. As I move down, this focus is before. Or as I move sideways as well. So if I look at the surface over which the rays cross as I move this, it would be a nice curved surface. What happens if I shift this to the other side? Yeah, so now when I move this, well, I can affect that by the focus in here. I can see that as I go to the edges, the focus moves in that direction. So we are down to field curvature. So 2 to 0, 2 to 2, 2 to 2. This is going back here. No, no, that's going to be different. Yeah, so we're going to get to that one. Yeah, it's distortion. Well, it's a spatial distortion, but it's not what we call distortion. Distortion is going to be, it's coming up. So distortion is a different thing. Distortion means that your image plane is still flat, but it's distorted. We'll get to it. So it's 2 down the road. We're almost there. So 2 to 2 first. This looks like, well, it's rho dot h squared. So it looks like an error in magnification, but that is linear in h. And what this does is very interesting. It introduces, it's called astigmatism, which is a slightly misleading name because we also use the word astigmatism for a related but different concept, which is what we have in our eyes when the curvature of our eyes is not uniform. This is astigmatism that even a rotationally symmetric element has when you're using it from the side. So when you're using it for what h equals 0, there's no astigmatism. But if you come to the side because you're looking at the lens like this, the curvature that induces on the field in this direction and in this direction are different. And that makes that, rather than having a sharp focus, you have a sharp focus in one direction and a sharp focus in another direction. So I can show you that here. It's a bit slow, this switch. So if I add some astigmatism, so that was 2 to 2, at the center nothing happens, but as I move off axis, I can see very clearly that all my rays are intersecting at a line that is at the radial directions. If I move to y, this line always points to the center. And then if I go a bit before or a bit after, the rays are crossing in the perpendicular direction somewhere else, sooner or later, depending on your... So here, if you have astigmatism, rather than having all the rays here, and at given points, you have that as you move away from the center, the folks are elongated, and if you move further away, they get longer, longer, longer. You can add a bit of field curvature to correct for that, and some people define astigmatism with a bit of field curvature, so that rather than the focus being at one of the two lines, it's halfway between the lines, and that gives you again the minimum confusion. But you can see here that if I were to focus this, then the line is in the other direction. So maybe it's easier if we show that here, if I add some astigmatism here, and I move to the side, let me add more astigmatism, the focus is a line, and it always goes pointing towards the center, but if I focus, there's an ideal plane at which this is nice and round, and then if I keep going, it's a line in the perpendicular direction. So it just breaks the rotational symmetry of the panel. Finally, we have W, or is it left one? 3, 1, 1. This thing is doing things on its own. Stop, stop. This one, 3, 1, 1, which I erase here, which is h dot h, h dot rho. This does what you were saying. So this one is the only one that does not change the fact that all the rays focus perfectly at the nominal plane. They just focus at the wrong place. And depending on the sign of this, if you were to image points on a grid, it would go to something like that, or something like that. This is called barrel distortion, and this is called pincushion distortion. So we've all seen this when we take a photo of a group with a not very good camera, and I've been often at the top right corner, and when it looks like this, that's distortion. So the people at the edges always stretch, and it's typically pincushion distortion. Nowadays that we have electronic cameras rather than film, is this one a big problem or not? Not so much because this one doesn't affect the quality of the image, just its shape. You can correct it electronically. You can remap your points and undo it. Yeah, so you would have a funny shape of image, and you had to interpolate and do things like that. Yeah, but other than these practical pixelization effects, it's not affecting the quality of the image. In addition, we interpolate using men's and statistic methods. Sure, sure, exactly. So while you cannot do that with the others. So this is one that... You could also, as I said before, with field curvature, you could make a CCD that is curved. Or you could make a CCD where your pixels are the form, so that you don't have these problems. Of course, that would be very expensive and not worth it. But it's something that you can, in theory, correct. There is a perfect image, it's just funny, stretched. And that's it. Those are all the third-order aberrations. So all the things that you can do to an image to make it less good. Of course, we have all the combinations, so it's easy to see what happens when you have only one. But when you combine things, it's harder to see. So just to finish this, let me show you here. I'll show you the same thing, but for an array of points. So, for example, this is not only one point, this is several points. So if I add distortion, this is much lower. This is pink cushion distortion, barrel distortion, sorry. Or pink cushion distortion, if I go the other direction. And you could add some stigmatism and some... What was this one? Steel curvature. Let me remove that one. That one's not very pretty. Some coma. And you can see how different parts of the image, different aberrations affect. But this point is perfect. What can I do to ruin that point? The focus on... But then I can move. But spherical. Spherical is the one that... If I have that, then... Oh, I put a lot of it. Even this point is not perfect. So aberration theory is just... It's very pedestrian, really. You just take this function, you say, I'm going to expand it into the simplest form I can. The top two terms are easy to understand. Then the five other terms, I see what they do. And I give them funny names. And then there are... When you have several elements and you concatenate them, these things add. And you can then try to compensate once with others so that you can cancel as much of them. I have seven minutes. That's probably what I need. This is all ray theory. Ray theory is very useful for getting a gut feeling of what's happening. But if we were going to do wave modeling of aberrations, how would I do it? What would be the way to model what with waves happens? Sorry? Yes. So what I can do is what we've been doing modeling and what I did with the first group yesterday in the mathematical workshop, simulating a system of this type. What you do is going from here to here, you use a Fourier transform. Then you apply a pupil. And then you can apply an e to the i wave front. So you multiply by a phase mask where the phase is e to the i, I guess k, the wave front, and then the wave front aberration as a function of h and rho. And then after that, you do another Fourier transform to get what you do. And that models nicely what these aberrations look with waves. Of course, the catch is you can only do it for a point source at a different location. You cannot do it for an extended object because this aberration depends on what point you're here. But when you go from here to here, you lose dependence on that point. So this is something that you can only do for point objects. But it still gives you a good idea of what the aberrations look like when you go like that. So let me see if I can do this here. So here, this is way too big. What I can do is... So this is the point spread function, wave optical point spread function. This is something that I haven't defined, which I will talk about tomorrow. It's called the MTF modulation transfer function for this system. And it happens that this is the perfect one. If you do focus, of course, your point spread function is not focused. It doesn't look good. Let me bring it back to focus. If I put spherical aberration, my MTF gets much worse. My dot got bigger. If I put more spherical aberration, it gets even bigger. And we can see a halo here with other things. Let me undo the spherical. Now I'm going to put... What happens if I do... I should say this is a point that is not at the center. So I'm fixing a point off-center so that I can see the effects of all the others. So then if I put some coma, that's the ice cream comb with wave. So I see a lot of interference effects. This is the characteristic 60-degree wedge here with the circular thing. If we put W2 to 0, what will happen? It's field curvature. So because I'm only at one point of axis, it will just look out of focus. So if I move that one, it's simply going to look out of focus just like before. So it's just out of focus because I could cancel that with a bit of 0 to 0 for that point. If I put this one, I get my astigmatism. The point spread function gets long. And I could combine a bit of the focus with that to make this more round. So correct it a little bit. It's hard to get right. That's probably the best I can do. And if I keep going the other direction, it's going to go flat in the other direction. And finally, what will happen when I add distortion? My point just moved because this corresponds to a point here. And when I add distortion, then it just moves down. It doesn't change the quality of the image. It just moves it to another place. So it's very easy to model the effect of aberrations with a web optically as well as ray optically. There are questions. How will be the image of some ring if my object is a ring due to aberration? How will it change? So if you have a ring, that's a good exercise. So let me erase this. If I have a ring, I'm imaging all points in a ring. Let's see what would the different aberrations do? So what would spherical do to this? Each point would get some circle of blur. So the whole ring would look uniformly blurred. So it would be uniform, but blurred. That's right. With spherical, yes. Then with coma as well, because each point is going to get its little ice cream cone here. So it would be blurred in a different direction. So there's only one that I think, only two are going to be special here. Field curvature, because it's a ring, it will all go the focus at the same time, assuming that the center of the field is here, because all points will be slightly the focus, but in the same amount. Distortion, what would happen to this ring with distortion? It would just get smaller or larger depending on that, because all the points are equally far from this. As long as, again, as this point is at the center of the image. If the center of the image is here, then it would get a bit oval or something. Astigmatism, what does astigmatism do to it? It will blur it a lot, because each point would go like this. However, if you combine the right amount of astigmatism and field curvature, each point would be blurred like this, and your ring would still be pretty sharp. So if all you want to image is a ring, and you have astigmatism, you can combine astigmatism with the focus or field curvature. Yes, to correction. It was my personal experience that I'd like to ask you. I have an experimental setup for practicing astigmatism. Thank you. Okay, very good. Any other questions? What change will you do in your program to change from point to ring? So extended objects for this program are difficult, because again, you have to write a different phase aberration on each point. For a ring, what I would do is I would just calculate it for one point and then do a rotational superposition of that that would make it faster. Because the other thing is what we're going to talk about tomorrow, too, is about spatial coherence. It's very different if this is a coherent image or an incoherent image. If this is a ring that you have a mask with a ring aperture and you illuminate that with a laser, that's very different that if you illuminate that with white light, let's say, or if you just have a bright ring. If it is a bright ring, then what you can do is calculate all these point-by-point functions for each point, calculate the intensity of that, and then add them all together. If you were to do it coherently, you would have to calculate the fields, not the intensity of each one of those, add them all together, then take them out to the square, and that would be it. Thank you. Any more questions? Maybe we should stop because there is a strict time, especially for people who have to go up. Yes. All right, so group two. We're going to be then next door. In group one.