# What are typical matches used for?

## Trouble in Science

Ladies and gentlemen, the subject of optical problems is a bit vague. What it is about is reacting to a long, long series of errors. I started writing a book about optics in the year 31. Like every physicist, like every professor, I felt the need to put down one of the many carefully prepared lectures that I have given in my life and repeated over and over in an improved form. And many have published theirs. In Germany, for example, Sommerfeld and Planck and others. Now in optics I've really finished and with two students I have created a rather thick book of around 700 pages and unfortunately that appeared just as Hitler appeared. And the success was that this book disappeared in Germany, so to speak. I don't think that much of that is in the hands of private individuals, the libraries will probably have it. I myself took a copy with me when I emigrated to Edinburgh and then didn't hear anything about it. I used it in my lectures there. At the end of the war I passed a bookstore and saw my book there, in a modified form with a modified cover, and found out that it was an American reprint with light photo technology. And then I made inquiries and found out: A few copies of the book have spread as far as America, where it was felt to be very useful for many optical and radar problems during the war, so the Custodian of Alien Property, the Handed over the manager of the foreign confiscated property to a company for reprint. And she sold it in large quantities. That annoyed me and I wrote to them and they wrote back that my matter would be dealt with at the right time, "in due time". But then nothing happened. A few years later I saw a note in the Manchester Guardian that the famous Finnish composer Sibelius had given a concert in New York - that was after the war - and that President Truman, who liked it very much, had spoken to him afterwards and asked how he liked it in America. Then he said, very good, just they confiscated all of my works and I don't have any more of them. Truman was very angry and wrote to the custodian in Washington that it must be okay and it was okay. Then I wrote a letter to the Manchester Guardian, where I explained that many others are in the same position, including myself. Then I got a letter from the custodian in which he wrote to me again that the matter would be dealt with - then nothing happened again. And then one day a scientific attaché from the American Embassy in London came to me and said that the Navy, that is, the American Navy, would like to ... that is a great patron and supports scientific institutes, that would like to publish an optical book and made a big plan and recruited many employees. Then by chance they would have heard that I was planning a book like this. And if this, my book, were to go with their intentions, they would step back from their plan, which I found very honorable. And right now I was ready to do a new book. It was no longer possible to re-publish my old book and so I wanted to do a new one, already had employees and presented my plans to him. This was approved and then things went smoothly. Because then the Navy attacked one that is apparently one of the most important institutions in the USA, and it only took me five years to get compensation and the rights to my book again. Well, that was good, I could have the old book reprinted now, but of course that is completely out of date. In the meantime, at the insistence of many English colleagues, I had started a new book and I succeeded in finding a colleague in a doctor, Emil Wolf, a young man who had been expelled from the Czech Republic, who was really professionally trained in optics in Cambridge. With this I have worked on this book for eight years and now it should appear. I can only hint at what happened in between. It really is so habent sua fata libelli, not only when they appear, but also before. For example, one thing was: One employee, we had about seven employees at the end, but before that many more who tried that, but it always didn't work. For example, someone delivered a manuscript that we had to reject entirely. And then he wrote very grim letters to the publisher and still asked for his fee. The publisher did not want to and there would be a lawsuit before we heard that it was superfluous because the man was jailed for fraud. It had nothing to do with us, but it freed us from that burden. Things like that, things like that, always happened. It was the first time that I directed a team like the one you have to do today. I myself contributed little to the book. I have read every word, all the corrections, I know it very well and it was written by Dr. Wolf. The employees delivered pieces of it, but he also rewrote most of it. He also brought out my old style, translated into English, so well that I can read it like my own writing. Then the book should appear now, I was hoping to give you a finished copy. I have one here too, but it's not quite finished. It is bound and looks the way it should be, but as you know, a printer strike broke out in England a fortnight ago. The only thing that wasn't finished was the subject index. So it has been fixed and impossible to finish for an indefinite period of time. Although I believe the Pergamon Press is trying hardest. And so they bound this book for me without the index. Otherwise it is completely finished, only the index is not in it. So, as you can see, it is quite a large structure. Now don't be afraid that I'm going to bring you this book on a larger scale, what I intend to do is to pick out a few small passages. But first I would like to say that this book, which is quite thick, is only a minimal part of what a real physicist calls optics. Already in the old book we left out, firstly, the optics of the very fast movements, the fast moving bodies, because that belongs to the theory of relativity today. We have also left out the optics of the creation, annihilation and scattering of light. Because that belongs in quantum theory. But we took something with us that was then called molecular optics - that is, a consideration of the movement of the light-emitting and absorbing atoms and molecules. That is, the temperature dependence of the optical effects. This chapter is about halfway in the old book. That is also missing in the new one. So everything that is really interesting for the physicist has been left out in the new one. So it's really more of a technical book. And yet I believe that it has a certain physical interest, because precisely these fine wave phenomena, which we are dealing with here, occur everywhere else in physics, in the wave theory of electrons and in other areas, and therefore the interest is not so pure is optical. But at least my employees were all real opticians. I'm not and so I directed a team of people who were completely different from myself. That was very instructive for me. But now I have said enough as an introduction and would now like to pick out a few things. One section for example ... First of all, what is really in the book is the propagation of light, viewed as a special case of the solutions to Maxwell's equations for continuously diffused matter, regardless of, or at least only, the atomic structure with very superficial consideration of these and the application of them to the optical instruments, of which I really know much less than of the things mentioned above. So that's essentially all done by my employees. But I learned a lot in the process and then I think I can fish out a few punch lines where I can tell you what this effect is actually based on. Now we have one of the great opticians among us, Professor Zernike, and Professor Zernike criticized my first book in the most gracious but severe manner, for which I am still very grateful. And my main interest now is to see that Mr. Zernike is convinced that the new book no longer contains the mistakes that were in the old one. The first thing I want to cover here, very briefly, is the theory of thin layers. For the reason that this is very important in practice. For example, by distilling or otherwise applying thin layers to a glass or other transparent surface, the reflectivity of this surface can be changed, strengthened or weakened as one wants. The whole problem, mathematically speaking, is this: you have a number of substances that are stacked in layers. Each layer has a certain optical refractive index and one wonders what kind of light is going out and how much light is reflected back? How much light goes through and how much is reflected? We owe the method that is used today to a French researcher, Monsieur Abelès. And he made a very good sketch of the matter for us after we worked up these others. But it was then new using the so-called matrix calculation. I have a blackboard here that I want to paint it on. So here are these layers, these red lines. And now a ray of light comes in here and then a distinction is made between two cases: a case in which the electrical vector oscillates parallel to the plane of incidence, i.e. in the direction normal to the wave and in the incident direction, and in the other case where the magnetic vector oscillates in this way. These two ... I just want to look at one of these cases. In any case, each of these two cases is completely independent of the other and behaves in such a way that it can be understood if the direction of propagation is here, which I want to call Z, and here, let's say the electric vector oscillates, and then here the magnetic vector in the Y direction. It's a right-angled system. Now I look at the light that passes one layer here before it goes in. And see what happens to the light when it reaches the next layer before it goes back in. Included in this transformation is first of all the passage through the interface, and secondly the propagation in the next medium. For every such process there are natural simple transformation formulas that ..., say, one component here, the electrical component U and the other V - that is the magnetic one here - convert into new ones with some coefficients A-B-C-D. And these coefficients can be calculated once and for all - in the magnetic case and in the electrical case, in these two cases. When you have that, the problem of treating many layers in a row comes out of it. This matrix is formed, that is, I assume that every physicist today knows what a matrix is from quantum mechanics. The system of these four numbers and the well-known multiplication rules. If we take this matrix A, then we have a matrix A1 for the transition from the first to the second medium. Another A2 and A3 for the transition from the second to the third, then to the fourth, etc. And now if we want to know what happens in the end. So we have to multiply the matrices according to the matrix control. This beautiful recipe was carried out there, by Mr. Abelès and also in America by a Mr. Billings and it is very powerful. I just want to show you a very simple example of how complicated things come out at the end. The first picture deals with the case that we only have three media. Let us say a glass substance on which there is a layer of another material, a thin layer, and above it is air or vacuum. So vacuum layer and glass. The thickness of the layer in wavelengths is plotted here. Here is a quarter wave length, half a wave length, three quarters, etc. From here on a different scale is used. That is why it is so disengaged. Up to this point is the scale - that is a unit - and here four tenths of a unit are only from here to then. So that is a very arbitrary unit. The first medium air has the refractive index 1 and the third on the other side glass, let's say 1.5. And the medium, the intermediate medium, can have any refractive index. And now the result appears as follows: If the intermediate medium has a refractive index that is much larger than 1, then this upper curve applies. And you can see that you can then vary. The reflections here are the starting point for the index of reflection - responsiveness, how much light is reflected. From 0.04 - very little - to a very high value: 0.5 for example. And this repeats itself periodically as the thickness increases. But if the thin, placed medium has a refractive index that is smaller than the larger of the two before, i.e. the glass, then it is the other way around, then it goes down, as this curve shows, but only tiny, because this scale is yes enormously enlarged down here. And I wanted to show you as an example how one can now, in a very simple way, if one can have a calculator sit down there and calculate all such examples. And it is the same in industry, where a lot is happening. I would now like to move on to an interesting problem. So there is the geometric look. This is the borderline case where the waves are viewed as so small that one can speak of rays instead of waves. Then there is a method of treatment that originally came from Willam Rowan Hamilton. But then later developed under the name Eikonal by Bruns in Germany, and which I took up in my first book in a form that it got from the astronomer Schwarzschild. This has caused a lot of trouble with the real opticians. Because it's very high in mathematics and they don't love it very much and attacked me, it was only about the simple one and it would be completely superfluous, and two parties have also formed. Hardly anyone followed that. How we were planning this new book, we were very open and said, maybe we will do it differently. We tried everything out with all the opticians available to us - it was a war back then - so we only had the English and American ones. And then it turned out that we all convinced them that our method is the best, this Schwarzschild method. So we went over to using this so-called Schwarzschild eikonal, which essentially consists of the fact that the wave is no longer regarded as a really vibrating wave, after all, the surfaces of the same phase in their progression are treated as wave surfaces and the equation that determines this , is called the eikonal equation because that is what Bruns called it. Now I would like to tell you that if an image is perfect, then a point of light, we have the light source Q here, again generated a point of light on the other side of the image surface by the imaging system. Point to point. This is the so-called absolute Gaussian mapping, which only exists in the roughest approximation. The geometric optics, alone, even without the waves, already delivers errors that come from the fact that the surfaces can never be constructed in such a way that this union really takes place. The theory of these errors was first fully developed by a man named Seidel and the lowest order errors are therefore also called Seidel's errors. There are five of them. Why five? So this is a general problem that I think every physicist should know. Why are there five mistakes? These errors are called spherical aberration, astigmatism, image curvature, distortion and coma. Now I want to tell you where it comes from that these five are. It comes from this: if you look at the entrance pupil of the instrument here, i.e. the opening where the light comes in, and then look at the image plane here, then you have a coordinate system in each and, of course, I can arbitrarily choose the point where the light comes from , on an axis, for example on the y-axis. And I call this distance Y0. Here, on the other hand, I can't do that, but here it will be somewhere and has an angle, which I called theta, and a distance Rho. So there are three variables. The distance Y0 of the image point from the center, then the distance the distance of the image point from the center in the image plane Rho and the angle that this is rotated, this ray that comes there. You now have to combine this to form a square combination, because you can easily see that the images naturally don't change when I change the sign of all sizes, so it has to be square. Therefore, you can form three invariant quantities, that is, quantities that do not change, if I just turn the whole instrument around its axis, then nothing should change. Hence, the invariants are essentially Y0 ^ 2, Rho ^ 2, and Y0-Rho-cos (theta). That is probably shown in the most elementary mathematics, these are the three invariants of two lines.However, six combinations of the second order can be formed from three quantities. Namely the three squares and the three products. The first combination is 17 and 0 ^ 4. Squared that again. And that's a given. So it has no descriptive meaning at all. That leaves 5 and these are the 5 image errors, so they are essentially Rho ^ 4 and then the square of this, to 0 ^ 2, Rho ^ 2, cosine ^ 22 theta, and then the products, there are five. The three products and two of the squares. But now, and this is one of the few points where I have contributed something positive in the book, I have found a method to clearly show what that means. Namely: I think this is where the beam arrives, goes through the instrument and unites here. So this is a concentric sphere that contracts from the point. And the sphere isn't exactly a sphere, it's a little bit distorted. The distortion of the sphere around a ray is now applied to the picture. And there you see 5 possible distortions and each time underneath it says what it is. So here you have spherical aberration, that works with Rho ^ 4, that's 2% of Rho ^ 2. It's such a flat, plate-like ellipse. That's the coma. That works with Rho * Rho ^ 2, Rho ^ 2 * Rho cos (theta), so that gives Rho ^ 3 cos (theta). That's a little crooked, not symmetrical. This is the astigmatism. That goes with cos (theta) and Rho ^ 2. That is the curvature of the field that goes with Rho ^ 2, so that is a paraboloid. And finally the distortion, which goes with Rho cos (Theta) and is like a cut through an ovaloid. I say these figures are new, they were absolutely not found in any other book and they give a very clear picture of what the distortion actually is in an instrument that has not been exactly corrected. I will come back to these image errors again later. At the moment I would like to move on to a completely different problem: Can't you make these image errors visible yourself? And this is what certain interferometers are used today. In other words, instruments that allow a bundle of light to be divided into two and then reunited again after the two parts have traveled different paths. That gear differences occur. The interferometer I am using here is a modification of the famous Michelson instrument, namely Twyman and Green. So I want to indicate the principle very quickly, you have a glass plate for separation, as with the Michelson, which, when hit by the light, reflects a part out and lets a part through. Here a collimator and here one and here a flat plate, a mirror. So the light arrives here, comes to the plate, is thrown up here, is reflected back here, goes through the plate and is reunited here by this collimator. Here at Michelson, however, there is also a plate, a flat plate. Instead of this flat plate, a very precisely spherically cut plate is now used and a very well corrected lens is placed in front of it. What happens then? No, you put the lens that you want to correct in front of it. That's how it is. Not yet corrected. The light that comes here, if the lens were corrected, would then in a sense seem to come from the center of this spherical mirror. If, however, the lens has defects, they are superimposed over this spherical surface as a small disturbance, so to speak. And that can be photographed, because the fact that the light subsequently reunites with the other undisturbed light that comes from here creates interferences that are an exact image of this phenomenon. For example, you can see above, observed, calculated below, the first image on the left is the spherical aberration, the second is the coma and the third is the astigmatism. I think the correspondence between the calculated and observed interference behavior is quite nice. They are tiny effects. Of course I do not accept this sheet. It was made by someone else, I wrote it down here too, but I don't want to bother you by name. In order to stay with interference, I would like to show you a nice photo. I want to mention the name, it comes from Professor Polanski, who deals a lot with diamonds and such crystals and is interested in what their surface structure is like. And for this he uses multiple interferences, that is, he takes the surface of the crystal and places a very well-cut flat plate over it. And then he lets light fall in and arranges it so that the light beam is reflected back and forth many times before it comes out. And as you know, the effect of elementary optics is that the interference disks become extremely fine. The more you interfere, the finer the stripes become. I don't want to tell you anything further about the finer theory of the matter, just want to show you one of his most beautiful pictures. This is a cleavage surface of a mica crystal, recorded in the green and in the yellow mercury line. What you see about it is that there is an irregularity running through the crystal through the surface, very clearly, some kind of crystalline cast or something. The order of magnitude that can be determined in this way is almost of the order of X-ray images. But I don't want to go into these any further, just show you what you can do today. Now I come to a very uncertain reason: I want to speak of the famous method of phase contrast by Mr. Zernike, who is here. Mr Zernike himself spoke to us about it three years ago, and I have to admit that I didn't understand very much about it at the time. Now, through the compulsion of the book, I have studied the matter a little and now I think I understand it in such a way that even someone who has no idea at all about optics can make it clear what it is about. There are two things to distinguish. For once a fairly trivial mathematical matter. If there is an object here and a microscope and the rays come through, then the object will have an effect on the rays that can be represented by saying that the light amplitude is multiplied by a factor f (x). O (x) here this direction, and one direction in the object has only one direction. This function f (x) is generally complex, that is, it means a distortion of the intensity and the phases. The shape, let's say: small f * e ^ i Phi. That is the amplitude and that is the phase. Instead, I just want to just write: f * 1 + iPhi + higher limbs by developing. The phases must be larger or smaller. Well, if I make a very thin section in the microscope, there is no noticeable change in intensity. Only with transparent substances, especially with light colored substances, this is f1. And when I calculate the intensity, that is, this squaring ..., or multiply with its conjugate, absolutely squaring, that's practically 1. You can see that here immediately, but also here 1. So you don't see it here at all. How do you make it so that you see what? This is based on a clever consideration of the nature of Abbe's microscopic imaging. According to Abbe, microscopic imaging consists in the object being illuminated here. And I want to imagine it as if the object were a grid. Then the first effect of the lighting is that a diffraction image is created here. First a continuous ray and then a much weaker first-order ray to both sides and then a second-order ray and so on. They form the first, second and third order diffraction images of such a grating and are completely analogous and generalized for any object. Then comes the imaging apparatus, which combines all these rays, or at least as much as it can grasp. If he unites them all, a similar image of the grid is created here, and if he omits some, a dissimilar image of the grid is created. That is the Abbe theory in very broad, crude traits. So he attributes the resolving power of a microscope to how many of the diffraction images the instrument can reunite. Well, what Mr. Zernike is doing is this, he says: If he could transform the 1 into an i in this function, it would be: i * (1 + Phi). And when I square that, it has a linear term. I ^ 2 is 1, but this gives 1 + 2Phi + square to i. Then there is only one i, of the kind you want to use it. So what do I have to do? I have to somehow achieve that this 1 turns into i in this permeability factor. But what does the 1 correspond to? It is very easy to see, that is the zero order continuous ray. And the phi corresponds to a higher order. Mr. Zernike, excuse me if I oversimplify it in your opinion, but it is as I believe that it may be pedagogically useful. Then you just have to do something in between, a body, a disk that changes the phase of the passing wave so that it becomes 1 i. Because the i means a phase of pi half a right angle. And you can do that. So you have a plate that changes the phase, only of the light passing through, but not of the diffraction images. And then you go from that function to this one, and you get the effect. It turns the phase effect into an amplitude effect. And that's the basic idea that we naturally built on, the exact exact theory can be found in the book. I would like to show you two pictures of such phase images. This is a piece of glass, and under A the direct image, where you can hardly see anything, only the boundaries of the piece of glass. Under B and C are phase contrast images with two different openings. Here you can still see a lot of details that cannot be seen here. So this is the success of this Zernikeean method. The next picture shows something organic, namely a frog epithelium. A and B are a direct image with two openings and C and D are a phase contrast image again with two openings. So in A and B you see extremely little. In C and D you can see a lot of details here, and here too, here they are exactly the opposite in strength, that of course depends on the randomness of the openings and something like that. But here you can see the extraordinarily greater resolution that can be obtained with this method. Now I come back to my image errors. If you had an ideal image, that is, a point of light united again into a point of light, the question is, how does it look like in terms of wave theory? Not just as rays that radiate from a point and reunite here, but as a wave that emanates from here, is then refracted, and here concentrates on a spherical wave, and how does such a spherical wave behave in the center under the assumption of course that this spherical wave is not closed, but limited by the diaphragms as in every instrument. The original theory of this dates back to 1885 and is from the German Lommel and it is still the best. Although Debye gave a different version of the matter with integrals in 1909, it gives a few new ideas, but Lommel's calculations are quite outstanding and must be used. So we did the math using Lommel's formulas, or rather taken from a work by Wolf and Linfoot, and I just want to show how cozy it looks near a focus. So where you think the light comes together nicely at one point. Instead, what really happens is a wild thing. So here is the ideal focus. The numbers that are written in there mean intensities. So that's very difficult to read for my eyes. Here it says 217, 477 ... 677. Then a lot of small numbers come around. So you see, there are a couple of high maxima here next to the focus, which of course is also a high maxima, and then smaller maxima here and a lot of other maxima around. This is what the picture looks like in reality. It is a very complicated mountain range. If you cut it in this way, in the focal plane in the focal plane here, then you get an image that you can read off here. In the middle a high mountain and accompanied by ever fading smaller mountain. This is the usual diffraction pattern. The spherical figure. A maximum and then the little tails on the sides. But here it is so three dimensional and it only makes one of many cuts that we have. A few of them are also shown in the book. So that would be the ideal Gaussian map. But what happens now ..., no, unfortunately my characters have gotten out of order. Before I discuss what happens if I now take the aberrations into account, one of our first collaborators, but who subsequently left, Professor Gabor in South Kensington at Imperial College, who published what is known as a method of reconstruction based on the following. To do this, I have to go back to this Abbe figure. It plays a role in electron optics. In electron optics you get into difficulties because the resolution of the instrument is limited by the ..., oh, I don't want to go into the technical issues, I just want to say what it is all about. You remember: the object sends out its diffraction figure, the diffraction figure is united by a lens, and when it is well combined - everything that is there creates a similar image. Gabor's idea was to split this process into two. Namely to bring a photographic plate here instead of the lens, and to photograph the diffraction image of the object first and then to illuminate the photograph again in the same way and then to reconstruct the image. If someone said that to me, that’s nonsense, it’s not possible. Because the diffraction image contains not only the intensities, but also the phases. The photographic plate does not care about the phases, but only about the intensity. So what you get now is a distorted image because the phases are wrong. The great feat that Gabor has achieved is to show, by looking over very narrow bundles, which are only those in electron microscopy, that very thin bundles can be arranged so that the phases simply have a negligible effect . And that you get good images by breaking the whole microscopic process into two parts. You first make a diffraction figure, and then you photograph the diffraction figure again and then get the object. Incidentally, the same idea was also used by Bragg in crystal theory, where the X-rays are recorded - they only give the diffraction image. And then he thought: Can't I continue to get the correct crystal image from the diffraction image through a second photograph? Of course, that doesn't work because it's wide-angle. But at least the Gaborian investigation showed what one would have to do if one wanted to do it technically. But I believe, as far as I know, it has never been fully carried out. So I would like to show you one of Gabor's examples. So here we have the original. There are a few names of opticians: Newton, Huygens, Young, Fresnel to Bohr. Here they photographed the diffraction pattern, which of course no longer bears any resemblance to the original. And this diffraction image is again sent through the same apparatus and then reproduces the original in a completely legible manner, so I can read it from here. Again Newton, Huygens, Young, Fresnel to Bohr. Of course there are small distortions, because the phases are of course unavoidable. I wanted to insert this here. That is also included in the book. And now, as the last point, I come back to the diffraction theory of aberrations. These five image defects that I mentioned and that I keep forgetting, which amused you very much, were calculated with ray optics, where you assume that the light consists of real rays that are refracted. Now what happens when there are waves? Then you have to take these rays and consider them as carriers of waves, taking into account the phases. Of course, according to Kirchhoff's theory of diffraction, this leads to very complicated integrals, and again it is Mr. Zernike to whom we owe the correct method. In the first version of my book, I developed this theory as best I could; within two years the whole book was ready. I only had a month to do this job. So I eagerly took the formulas as they were and tried to evaluate integrals. That didn't make much good, but it did give an insight into the matter. Later it was recorded by different people, by Blaser and, of course, by Zernike. And Mr. Zernike found the right method. The correct method is to develop the waves according to certain functions, which we now call Zernike polynomials, and these are polynomials, that is, XY multiplied with each other and provided with factors and added again. And these polynomials have this property: They are defined in circles. The circle, the opening circle of the instrument ... and if such a polynomial is taken in the original coordinate system XY and this coordinate system is rotated into a red ... X ', Y' - so you can of course X 'and Y', X , Y and the pivot point in principle express here, in the usual way. The dashes of the X cosine and Y sine here etc. And then the new polynomials that arise from the old one should be identical to the old one, apart from a factor f, which only depends on the angle of rotation.As a result, the polynomials are essentially uniquely determined, that is also normalization, and, as Mr. Zernike has shown, they are the right elements on which to build this theory. Of course, there is no point in going into the theory itself here. I just want to show how now, when you take photographs of instruments, of lens systems, which still have the various errors that have not yet been completely corrected. How the picture looks like in terms of wave theory. So I have three pictures. The first relates to the meridian plane, that is, a plane that contains the ray itself. And this ray is so or contains the axis of the instrument. The first is spherical aberration. The second and third relate to planes that are perpendicular to the axis of the instrument. So that's how. And the first relates to coma and the last to astigmatism. So these are at the same time generalizations of my second figure, where I showed you these small areas that represent the geometric distortion and at the same time the image of an ideal bundle just shown in focus. If you now superimpose these two things, the geometric error with the wave nature, such figures arise where the isophotes, that is, the lines of equal brightness, are entered here, here is 75, goes down, here is 5, 2, 1, here it goes back up to 9. So, as you can see, the pictures are getting very tangled. This, like I said, is an image of the spherical aberration in the meridian plane. So the light propagates in this direction. This continuous line is the so-called Gaussian line of geometric optics, which I will not go into. You have calculated three theoretical figures above, according to the formulas, for the coma, which anyone who operates optics here knows that this strange figure is so distorted to one side. This has to do with the fact that the small area was distorted so crookedly. Above are the theoretical pictures for different cases and here below are the corresponding experimental pictures. And if you compare this picture, with the theoretical one you can see how it agrees nicely here or they are not so precisely assigned, but just to show how well experiments and theory agree with each other. The last picture, that shows the same thing for astigmatism, but I don't want to go into details. The book then contains the exact theory of diffraction for the few cases where it is possible. That means, for spheres and for half-planes. When you have a diaphragm made up of half a plane and the light falls in, what becomes of it. That is one of the famous problems that Sommerfeld first solved. But today's mathematicians treat this very differently. Not with such potentials as Sommerfeld did, but directly with so-called dual integral equations. But I can't bother them with it either. These are just methodological questions. The picture that I want to show you here only shows what it looks like when you really calculate these Sommerfeld formulas. He never did that himself. And here in one picture I have the phases and in the other picture the intensities. And indeed it is like that, the light ..., here the shadow is evident, nothing happens here. Here the lines are of the same intensity and here the same phase. So you can see how the shadow formation is created here. In a sense, these are the rays that are bent a little upwards. Here is the reflection, the light falling here on the shadow against the body is thrown up and interferes with the incoming light and gives such interference. Here is the same for the phases. Very complicated figures are created here. This is only to show how one can really calculate Sommerfeld's formulas in every detail today. The last picture I want to show relates to a theory originally devised by the French physicist Brillouin who predicted the phenomenon, namely: Imagine a vessel with a liquid in which very short sound waves are generated what you can do electronically today. We think of these sound waves as going from the bottom up. Then these sound waves are places where the refractive index changes - changes periodically. So you are a grid. And if I now let a beam of light pass perpendicularly to it, it will be bent. That sounds like a nice gimmick, but it's much more than that. This is because so far it is probably the best, and perhaps the only way to observe optical properties for small bodies. I can still make such very short sound waves even in crystal mirrors that I can hardly see. With it I can determine how fast the light goes through - from the differences that I see. And so I can calculate back to the elastic coefficients of the crystal. In this way, the elastic constants of rare crystals are known very precisely today. But here I only want to carry out the theory for liquids and not for crystals. There were a lot of approaches there, from Wannier and others, and then from the Indian, Sir C.V. Raman, who was also here three years ago. But who only wanted to deal with a distant borderline case ... could. At that time in Edinburgh he was a Canadian Noble and an Indian Bhatia, and I brought them together and suggested the method to them, and they then carried out the theory. Since it was about this, I want to record it again very briefly. Here is the vessel, down here is the crystal that generates vibrations, and here the sound wave runs upwards. It does not matter whether that is a standing sound wave, by reflecting it down, or a moving one. The sound is so slow compared to the light that that ... that a moving wave stands still and then you have a light source here and an imaging instrument here, with a parallel beam of light when it comes in here and then again through a lens is united at the focal plane. This is where the interference occurs, so it acts like a grid through which it goes. The angle of incidence has to be very small, otherwise nothing can be seen. The method we used was obvious to me because it is essentially the old perturbation method that was introduced into quantum mechanics in the theory of Heisenberg and Jordan and myself. With this perturbation method, which is a very strange case of degeneration, the two succeeded in calculating in this way, and as a result I would now like to show you this picture. Up here is an experimental shot of the Indian Partasariti. And what you can see most clearly there is the asymmetry on the right and left. Although it looks like it has to be very symmetrical. If you think about it carefully, it is not symmetrical, because the beam has a very small angle of inclination of 2, 3 ° at most. But this tiny difference means that, for example, only three lines appear here above and five below. And here the difference is even bigger. And, as you can see, these intensities are very intricate. And here the angle of incidence is given up here. It's hard to read: zero, 0.06, etc ... degrees! And now here it is stated: F1 and M2 are the number of straight and non-straight lines, one to the right and the other to the left of the center of the straight through ray. And there you see the same dissymmetry. Here it is still fairly symmetrical, with a very small angle, here it starts to become very asymmetrical and then, strangely enough, it becomes symmetrical again. So this is not a simple law where it changes from an asymmetry back to a symmetry. So that worked out well in theory. The theory is given here in brackets and the observation without brackets. so it is not an absolute match, but of course it is also an estimate of what one still wants to see as visible. After all, we were quite happy with how we had it. In any case, we have all the borderline cases that existed in the literature and can now subsume them. Here is a brief overview of this book. I would just like to add one more thing, this is a very important chapter: We have only ever considered the light source as a shining point. But the light sources are extensive. This expansion causes the phenomenon of partial coherence to occur, i.e. parts of the light source with other parts no longer vibrate independently of one another, but are instead coupled to one another. There is a very large chapter about it that comes from my colleague Wolf, and in which Zernike's results have again been used to a great extent. The book also contains some appendices, one of which I wrote in full myself. There is a generalization of geometrical optics, the so-called calculus of variations of mathematicians, and I have presented it to show that the essential optical phenomena take place in the same way to a much greater extent if one takes more complicated functions instead of the refractive index as the only characteristic quantity, that characterize the substances. This is necessary when you operate electromagnetic optics, i.e. electron microscopes. That is also in an appendix by Dr. Gabor has been treated. Finally, let me show you this is what it looks like today, apart from the index.

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