The lecture is devoted to the relevance of the Fourier transformation in science. It is fundamental to any experimental exploration where input-output relations are being exploited. Experimental results, obtained from a time-domain experiment, need to be transformed into the frequency domain for comprehension; and data from momentum space or k-space investigations require a transformation into the geometric space for visualizing the results.

Applications are plentiful. The first usage happened in optical interferometry, starting with the research by A. A. Michelson. Later, magnetic resonance, in particular NMR profited enormously from applications of the Fourier transformation. Imaging procedures, using x-rays or magnetic resonance for visualizing molecular structures and the interior of macroscopic objects are today among the most prominent applications of the Fourier transformation. Undoubtedly, clinical imaging in human medicine has profited most significantly from this simple mathematical transformation. A survey on these exciting consequences of the ideas of a great mathematician, who acted at the same time as a politician, is presented.

- Mediatheque
- Meetings

Dear friends, I'm enormously enjoying this year's Lindau meeting. Especially talking to you, my dear young enthusiastic and promising scientist friends. Among the lectures, so far, I particularly liked those that went beyond traditional science and revealed also the societal context. And in fact last year, at the same place here, I was talking about academic opportunities for conceiving and shaping our future. And in this context I made some rather strong and perhaps even offensive statements, for example condemning egoism as the driving force of all our actions, where we always ask ourselves what do we gain back from doing something. And rather promoting responsibility as the driving force, where we ask: What can we do in order that society profits something of it? And my lecture ended then with two quotes, one from François Rabelais: Or "Science without conscience ruins the soul." And on the other side, despite all the misery in our world, we have to remain optimist because we all together are jointly responsible for what will come in the future, we can't blame anybody else. So today I don't want to make any offensive statements and I will try to be a good boy and just tell you a small story, a purely scientific story: "From Monsieur Fourier to medical imaging." And in fact I like to demonstrate to you Fourier Transformation as a beautiful example how useful mathematics can be in the sciences. But, as usual, the inventor of Fourier Transformation, Monsieur Jean-Baptist-Joseph Fourier, he didn't know that today, at the beginning of a lecture, you could even peek into the head of the speaker in order to verify whether it's worthwhile to listen to his lecture, because all I can tell you is contained in this little sloppy piece of tissue here. So a lot of development went on from here to there. And I'd like to ask the question: Why is the Fourier Transformation so important in science? It's a very simple expression, it's an integral transform where we transform a function in time domain into a function in frequency domain. So we relate these two domains here by this integral transform. Or we can relate a momentum space to the geometric space, a function of K, of momentum and related to a function of coordinates. It has something to do with exploration of nature in general. When we explore an object, we consider it as a black box, we don't know what is inside and we try to perturb it. We knock at the input here and listen to the output. And that tells us what is inside. And actually a very unnatural way of exploring a black box is to apply a sine wave, a repetitive perturbation, and then listening to the output here. And I mean, just remember the lecture of Professor Hänsch two days ago, he told you that never measure anything but frequency. And that's exactly what we try to do in order to characterise the inside of this black box. And that makes sense because actually these trigonometric functions, they are Eigenfunctions of linear time-invariant systems. So whenever we apply such a function to a black box, which is linear and time-invariant, we get the same function back, multiplied with a certain complex quantity, which is, in quantum mechanical terms, the Eigenvalue, the Eigenfunction multiplied by the Eigenvalue. And if we block this Eigenvalue as a function of frequency, if we vary the frequency, we obtain the spectrum, that's the basis of spectroscopy, so it really makes sense. Now, we can do spectroscopy just in the normal way, applying one frequency after the other and measuring the response and blocking the amplitudes here as a function of frequency, that gives us a spectrum. But we could also do it in parallel, apply all these frequencies at once to the black box and saving time. Then we need something like a frequency sorter which sorts us out the various responses in order to again get the spectrum. And this frequency sort, that's after all nothing else than a Fourier Transformation. And by doing it in parallel here, we gain by the Multiplex Advantage, having done everything at the same time, often called also the Fellgett's Advantage. That's the advantage of going this way here. And the secret of the Fourier Transformation actually is also the orthogonality of the trigonometric function, when you multiple one with the other and integrate over this entire space, then you only get something when the two functions are identical. So they are orthogonal. And this allows one to separate them. And the simultaneous application of all these frequencies that can for example be implemented by a pulse. If you are by a pulse, then you have essentially all the frequencies contained, you obtain an impulse response. You have to do a Fourier Transformation and you obtain a spectrum, so simple. So again we have, so to say, two spaces here, which we relate, conjugate variables, the time and the frequency variable. The momentum space and the coordinate space, which are connected. There are, so to say, two different views of the same object. You can look at that in red colour or in green colour, and Heisenberg has said that a long time ago, that the most fruitful developments have happened whenever two different kinds of thinking were meeting. So these are the two different spaces. Or I mean, Professor Glauber, he has been telling you about waves and particles, this relation also belongs to the same category. And he also mentioned Niels Bohr, "Contraria sunt complementa", and he has here the yin yang symbol, this also represents this two spaces which, so to say, contains the same truth. The whole world of Fourier transforms in spectroscopy is like complex. There are many different possibilities. In particular because actually what we are looking at is not just a function of time or a function of momentum, but it's set both at the same time, depending on four different variables, it is a plain wave, which develops in time and it develops in space. And so we can either use the time dependence, make a time domain experiment and finally obtain a spectrum. We can also use the K variables, the momentum variable and obtain, here for example the image of a molecule, that's the x-ray diffraction. We can do imaging, magnetic resonance imaging, which also uses this K dependence as the Fourier transform obtains an image of a head. And finally we can do for example interferometry, where we use the R dependence, we measure the interference in spatial domain, Fourier transform it and obtain again a spectrum, for example an infrared spectrum. So these are these various possibilities, which I briefly would like to describe. And everything goes back to this gentleman Monsieur Fourier. Who was Monsieur Fourier? That's him. He was at the same time, and that's a very, very great exception, at the same time a scientist and a politician. I mean, there are very, very few politicians who understand anything of science and there are very few scientists who are interested in politics. But he is, so to say, my role model, he has the same in him combined. And even, I mean, here he is in his office in Grenoble, and instead of studying his legal papers, he is doing a physics experiment, he actually measures here heat conduction, and he was writing at the same time a paper, which he presented actually at the Institute de France, being prefect to the département Isère. He was writing a book, 1822, on theory of, théorie analytique de la chaleur, heat conduction. And he did experiments and also this kind of input-output experiments. He had a black box here, a blue box, applying heat from the left side and measuring then the temperature distribution in the body. So it's again this input-output relation, he expanded the input in the Fourier Series and reconstituted the result again from a Spatial Fourier Series. So he used for the first time this Fourier expressions, of course he didn't call them Fourier expressions, but anyway, here they are in his book of 1822. Whenever somebody claims to have discovered something, of course it's interesting to ask what has developed out of that, but you also have to ask who was before, and there is always somebody before. And very few inventors really invented something for the first time. So for example Leonhard Euler, if you go through the text books of Fourier transform, you know you'll discover the Euler equations for the Fourier coefficients. These equations here, which I showed before, these are called Euler equations, so Euler must have contributed something. And indeed he did that already in 1777, and even before in 1729 he used trigonometric functions for interpolating functions and that's a reason why he's here on a Swiss bill, obviously he was Swiss, and otherwise I wouldn't speak about this to you. So indeed the Fourier transform is a Swiss invention, keep that in mind. So, I mean, we should come back to spectroscopy now. I mean, there is a huge range of different frequencies, from the gamma rays to the radio frequency and you can use all of them to do the spectroscopic explorations of nature. And you have the tree of knowledge and you like to go from physics to chemistry to biology and medicine, of course this is the most important level here for the true understanding of nature. Whenever you can explain a medical phenomenon in terms of chemical reactions, then you understand it, you normally don't have to go back to physics. That's the level. But I mean, in order to climb on the tree and come safely down again you need a tool, you need a ladder, spectroscopy serves that purpose. Anyway, the first time something like that, in connection with Fourier Transformations, has been used, was this gentleman, Michelson. He has also been mentioned in the lecture by Professor Hänsch, and Michelson, he invented the interferometer, so he has two light beams, or essentially one light beam, which is split in two parts, one part going to this moveable mirror and the other to the static mirror, coming back, being combined again. And there is an interference occurring here, so that's the machine as it happens here. Let's look at it in a little bit more detail. So again, the incoming wave here being split into a blue part, a red part and they are coming back and here, in this particular case, the two waves being reflected from the two mirrors are in phase, so there is constructive interference. If you move now this mirror slightly, then you get a destructive interference, the two are out of phase and if you add them you get zero. So in this way you can actually get an interference pattern. And that leads to interferometry, and when you have a single frequency, then you get just a single oscillation. If you have two frequencies present at the same time, you get this kind of interference here, between two frequencies, two frequencies with a certain line shape here, you get an attenuation and recovery and you have all these different signals which, when I looked at them first, I thought they are free induction decays from NMR, looks very simile to NMR spectroscopy, but it was done 50 years earlier. Anyway, a modern interferometer works exactly the same way, the source, the sample, which measures absorption or tries to measure absorption. A translucent mirror, mirrors here which reflect a detector value records interference as a function of the placement of the mirror here, to a Fourier transformation and gets a spectrum. The very first time this has been done was in 1951 by Fellgett, therefore the Fellgett advantage. And here the interferogram, here the Fourier transform of it, he had to do it by hand because he didn't have a computer at that time. But today you can buy these commercial instruments and they do the Fourier transformation automatically, and you get beautiful spectra without having to understand what is going on inside of the box. Something similar you can also do with Raman spectroscopy, you know in Raman you irradiate with a single frequency, a laser for example, you measure this captive light here and the frequencies are modified by the internal vibrations of a molecule for example, giving you this additional, so to say, side band of the same frequency here. And this contains now virtually the same as an infrared spectrum. That's a typical Raman, Fourier transform Raman spectrometer, where the same principle is being used. And it just gives me a chance here to tell you something about my passions. And to tell you how important passions are. I use this kind of Fourier Raman spectroscopy for the pigment analysis in central Asian paintings which I have a great love for. For example here you have Raman spectra of different blue pigments and you see just how different they are, you don't have to understand them, you just see they are different and this way you can distinguish indigo from azurite, from smalte and prussian blue and you can now get inside of paintings and identify the pigments. I'm doing painting restoration, so it's important to know what the artist has been using. And in this way you can, without destroying the painting, you can analyse pigments, fascinating. You see, when you want to walk along this road here, your professional road, for example towards Stockholm, then, oh it's so difficult to walk on one single food, you need a second one and the second one, that's your passions. And only when you have passions in addition to science, or whatever you are doing, then this spark will appear in your brain and the creativity occurs, that cross talk between the two legs is very important, keep that in mind. So let's come now to NMR, another application where the interference now happens in time domain. Very simple experiment, you have a nucleus which is recessing in a magnetic field, nuclear magnetic moment, having a frequency being proportion to the applied magnetic field, so in essence measuring the frequencies, you measure local magnetic fields, that has been done the first time in condensed matter by Edward Purcell and Felix Bloch, you can record spectra, like here of alcohol you get three different lines, because the local magnetic fields in the methyl groups, the methylene group and the OH proton here, they are different, so you can distinguish. But it's tedious to record one line after the other, it takes more time than I have for my lecture. So we had an associate in Palo Alto, there was Wes Anderson, he said: "Why not do it in parallel?", invented the multichannel spectrometer, where he irradiated with several frequencies at the same time. He built a multi frequency generator, this so called Prayer Wheel, it never worked, it's now in the Smithsonian museum, but at the same time he had a Swiss slave in his lab. And together with this Swiss slave they thought: ah, it's very easy, I told you everything before. Just apply a pulse, observe a free induction decay, do a Fourier transformation and you have a spectrum in fractions of a second. Here the impulse response of these molecules, the spectrum, and here the spectrum which you would have recorded with a traditional sweep method, the snail crawling through. Anyway, simultaneous excitation leads to sensitivity and you get beautiful spectra. And we felt great, at that time we published it, we thought we were inventors and we didn't know that before Mr. Morozov, he did very similar experiment about six years earlier. Fortunately the committee in Stockholm couldn't read Russian, otherwise you would have to listen now to a lecture in Russian. But anyway, he didn't know why he would do this crazy experiment, I mean, free induction decay, the Fourier transform of it, he didn't recognise that it could gain sensitivity this way, and really shorten the experiment time, I don't know for what reason actually he did it. Anyway, he was the first. So we have now this beautiful spectra, but they are virtually useless. How do I interpret all these lines, you remember Kurt Wüthrich's beautiful lecture, he wanted three-dimensional structures of molecules, and we were working together at that time in Zurich, and so he wanted to go from a primary protein structure to a three-dimensional structure, and the question was how. You need additional information, for example you need this correlation information, you have to relate nuclei, how near together are nuclei in space, how near together are they in the chemical bonding network. And this kind of information gives you really geometric information to get the structure. And so that leads to a correlation diagram where you correlate different nuclei and that could be neighbourhood in space, neighbourhood in chemical bonds, for example nucleus G has something in common with nucleus A, nucleus F has something in common with nucleus C, that leads to two-dimensional spectroscopy. Here all these correlations are being displayed, you can use them to determine structures. And the idea for that goes back to Jean Jeener, he proposed this kind of two-pulse experiment, bang two times on your black box, and in essence you transfer coherence from one mode, one transition in the energy level diagram to another transition in the energy level diagram. And that tells you something about connectivity of the nuclei. And this allows you to get this correlation or cosy spectra here from the Wüthrich group, which allows you then to make assignments of the protons, for example along a poly peptide chain. You need an additional experiment, you need also the through space interactions for this, you use a three-pulse experiment, first again some blue frequencies which are being transformed into red frequency, but here through close relaxations, through the space, depending on the dipolar interaction, so you really can measure distances. That's a complete experiment, that's a two-dimension cosy, a nosey spectrum, you measure the distances here between neighbouring amino acid residues, you can get then the complete set of information, chain coupling information, dipolar couplings, you can make an assignment, false resonance and finally determine geometry. And you are in business. That's the first example which Wüthrich was also mentioning three days ago. And that's why he got his Nobel Prize for this ingenuous technique how to determine three-dimensional structures of biomolecules. Nowadays, when it's going to larger and larger molecules, inventing more and more tricks doing three-dimensional spectroscopy, doing four-dimensional spectroscopy, unfortunately I can't demonstrate the four-dimensional spectrum on this two-dimensional screen. But anyway, pulse sequence is becoming enormously complex, it's like a score of a symphony orchestra, you see the first violin, the second violin, the violas, the cellos and the percussion down here. That's the kind of pulse sequence which we use today, and you say, oh that's much too complicated for me. But even 10 years ago, when you wanted to find a job in industry, Merck research laboratories, you had to have experience in modern 2D, 3D and 4D heteronuclear NMR, otherwise you just were not considered. And today, Wüthrich told you, go up to seven-dimensional spectroscopy, that's important for finding jobs. NMR is also a beautiful example for determining molecular dynamics features. While x-ray diffraction delivers you the most reliable structures of biomolecules. NMR allows you also to go into dynamics and see what happens in a dynamic molecule and, I mean, static molecules, they're so boring, they are dead. Life is dynamics, dynamical molecules, that's where really is interesting chemical reactions, interactions with molecules, for that NMR is a beautiful technique. You have here an example, you have a benzene ring with seven methyl groups attached, you want too much, you think normally there are only places for five substituents, so methyl group number 7 is being chased back and forth here between the different positions and the question is how does it go? Is this methyl group jumping just to the next position step by step, or can it jump also directly into position? For what is a network of exchanges in such a molecule? To just record a two-dimensional spectrum, you have the four resonances here, 1, 2, 3 and 4. You have cross peaks and these cross peaks tell you how the jumps go. For example, if there would be a random exchange between all positions, there would have to be cross peaks everywhere. If there is only a 1-2 bond shift, then these are the circles which you prefer then, you see it fits. So indeed you have immediately determined the mechanism, how this exchange goes, you don't have to understand two-dimensional spectroscopy, it's a beautiful display, tells you everything. I mean, NMR is on the very far end of the spectrum, low frequency, there are other spectra, EPR, microwave spectroscopy, coherent optics, for example. And exactly the same principles apply here also, except that the practical difficulties increase, going to higher frequency. Electron spin resonance is perhaps the most similar technique to NMR, where just an electron spin is coupled to many nuclei, giving a multiplet here, very complicated spectrum which one can analyse with traditional methods or with Fourier methods. If the spectrum is very narrow, like for organic radicals here, one can really use directly the Fourier techniques. For transition metal complexes, the spectra are enormously wide and you cannot cover it with a single radio frequency pulse. But for this organic radicals you can record again an impulse response of free induction decay to the Fourier transform and get the spectrum. Exactly the same, you can get two-dimensional electron spin resonance spectra, so the same principles apply here as in NMR. When you have broad spectra, then you have to do more specialised experiments, I don't have the time to go into that, it goes into endopulse, endoexperiments, I don't want to describe that here, you measure then a modulation of an echo decay, do a Fourier transformation and then an ENDOR, an indirect detected NMR spectrum, but I don't have the time to explain that. And if you want to know more about that you can read this book by Arthur Schweiger and Gunnar Jeschke, Arthur Schweiger unfortunately just died two or three months ago, being less than 50 years old. Anyway, that's his legacy here, you can read about this beautiful experiment. Then, in the same frequency domain, in microwave spectroscopy, you can also do rotational spectroscopy, where molecules are rotating and you're measuring the speed of rotation about different axis, also internal rotations, that's microwave spectroscopy in the true sense, Flygare did the first pulse experiments, you see the free induction decays, you see the Fourier transform of single lines here, so to say, or single multiplets. You can go inside here, determine high resolution spectra, and making particular assignments, assignments of resonance lines which have for example one energy level in common, so this red line and this red line, they must give a cross peak here, somewhere, which tells you that, so you can study connectivity in energy level diagrams by these kind of two-dimensional spectroscopy. And then you can also go to optics, to optical time domain experiments, optical pulses, there are very short pulses, there are picosecond pulses, femtosecond pulses, done in the lab by Robin Hochstrasser here, it's a 4-pulse experiment, to study actually chemical exchange in real time, so to say in a biomolecule. And here the beautiful results, 2-dimensional optical spectra. So again exactly the same principle, it's just a little bit more tricky and more difficult, but gives you this beautiful spectra, which I don't want to interpret. You can apply exactly the same principle to mass spectroscopy. You can do time-resolved experiments here in an ion cyclotron, that's a magnetic field here again. You shoot in ions here, they start to circle around, you excite them by a radio frequency pulse and you measure again a free induction decay, here in mass spectroscopy. And you can for example distinguish here between two ions which have virtually the same mass, there is a very slow interference pattern which you can analyse by a Fourier transformation. You get very highly resolved mass spectra, but you can do that also for complicated molecules, here for a protein or a protein complex actually, which you can investigate by Fourier transform mass spectroscopy. So you see it's the same principle, it's virtually always the same, and it goes on and on and on. Then you can also do diffraction experiments, I mention this dependence on K here, Fourier transforming into real space determines this shape of a molecule. That leads to x-ray diffraction. Again you measure here structure factures in K space and the reciprocal lattice, you fully transform, you get electron densities in geometric space. Again it's the same kind of principles which apply here, here from a book, from x-ray diffraction, you see exactly the same kind of expressions here also occurring. An example in myoglobin, in the background you see the diffractogram and the Fourier transform structure here in front. I mean, you know this beautiful example of Michel, Deisenhofer and Huber, Professor Huber will probably speak about similar subjects this morning. Photosynthetic reaction centre being determined in this way, all relies on Fourier transformation. And finally I am coming to the last possibility, namely imaging. Imaging where you do an experiment which is very similar to diffraction. But you do it here in a slightly different way, you do it with magnetic resonance, with NMR, and you can in this way peek inside into the body. For example of your boyfriend, if you want to marry him, at first put him into a magnet and see what is wrong inside, whether he has a strong spine, whether there is anything in his head still left, whether he has soft knees, all that you can find out from MRI, Magnetic Resonance Imaging. And of course there are two windows to peak into a human body, you can use the x-ray window, you can use the radio frequency window, with optical radiation it's difficult to see through. But these are the two windows available. But the problem with magnetic resonance is resolution. How do you get with this long radio frequency waves spatial resolution? And the secret has been proposed by Paul Lauterbur, he said: as here, the nuclei have low recession frequencies and here they have high recession frequencies, so you get spatial resolution". That's what he got his Nobel Prize for, 2003. Applying magnetic field gradients in different directions, getting, so to say, projections of the proton density here, along different directions. And then from this projection one can reconstruct an image, that was his procedure. And the first time I heard about that was at the conference in the United States, 1974, and he showed an image of a mouse. It was recorded it in this way here, the mouse. Here these are the lungs, I mean it's proton imaging. And again, going back to what Professor Hänsch told you two days ago, never measure anything but hydrogen, that's exactly what we do in imaging, using protons for imaging. But there must be protons here in the centre, but what is that here? Nobody could understand what this feature here is. So somebody had the brilliant idea, this must be the soul of the mouse. But then, unfortunately, this poor mouse died in the magnet because the experiment lasted for such a long time. So then one found out, it's just an imaging artefact. So I got another idea, use the Fourier principle, apply to it in sequence first a vertical field gradient, then a horizontal and combining it to a dimensional experiment, do a Fourier transform and you get the image of a head. Data Fourier transformed in two dimensions, and that reveals everything. For example if there would be a tumour in my head, you would see it, fortunately it's not my head. That's an important image, that shows you how you convert a female brain into Swiss cheese, just drink too much. And you see the female brain, a normal female brain, an alcoholic female brain, who wants such a brain, so stop drinking alcohol, especially if you're a female, for the males it's less dangerous. Anyway, that's all you have to remember from my lecture and that's very worthwhile, small glass on this side, big glass on this side. And I mean, I know I should stop, I could go on forever, you can measure angiograms, blood vessels, you can look at chemical compositions after a stroke at different parts of the brain. You can do time resolved spectroscopy, Peter Mansfield got his Nobel prize for that, at the same time with Paul Lauterbur, using a particular pulse sequence, getting movies of a heart motion. And finally you can look into the brain and see what is going on while you are thinking, if you are thinking. And there is a particular principle which allows one to make NMR sensitive to thinking processes which I can't explain. You can get beautiful images here to distinguish a normal person from a schizophrenic person when you apply a certain input paradigm to him or her. And if you see a particular reaction, you know he might be ill. You can explore for example even compassion, that a person who suffers pain being tortured and you are just an onlooker and you feel then a reaction in the brain at exactly the same place as this person which is tortured himself, just by compassion. So it's quite exciting what you can do and I'm sure I have proved in this way that Magnetic Resonance Imaging is an irrefutable testimonial to the enormous value of basic research, it's directly linked to practical application. And finally: Happiness is finding still another use for Fourier Transformation. Thank you for your attention.

# Richard Ernst (2006)

## Fourier Methods in Spectroscopy. From Monsieur Fourier to Medical Imaging

# Richard Ernst (2006)

## Fourier Methods in Spectroscopy. From Monsieur Fourier to Medical Imaging

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