Ladies and gentlemen,
The topic of optical problems is rather vague.
It is a matter of letting off steam after a long, long series of errors.
In 1931 I started writing a book about optics.
Like every physicist, like every professor,
I felt the need to record in lasting form one of the many carefully prepared lectures
that I had given in my life and regularly repeated in an improved form.
And many have published such material.
In Germany, for example, Sommerfeld and Planck and others.
Now, in the field of optics I actually managed to complete it, with the help of two students, a thick book of about 700 pages
which unfortunately appeared just when Hitler also appeared.
And the effect was that this book disappeared in Germany, so to speak.
I don’t believe that there are many in the hands of private individuals, though the libraries should have it.
I myself brought a copy with me when I emigrated to Edinburgh and then heard nothing more about it.
I used it there in my lectures.
At the end of the war, I once went past a bookshop and saw my book standing there, in a different form and with a different cover,
and discovered that it was an American reprint using the photo offset process.
And then I looked into it and discovered this: A few copies of the book had spread as far as America
where, during the war, it was found to be very useful for many different optical and also radar problems,
which resulted in its being delivered to a firm for reprinting by the Custodian of Alien Property
which handled foreign confiscated property.
And they sold it in large quantities.
That annoyed me and I wrote to them and they wrote back saying that my business would be attended to "in due time".
But then nothing at all happened.
A few years later I saw in the Manchester Guardian a notice of a concert given by the famous Finnish composer Sibelius in New York
- the war was already over – and that President Truman liked it very much and spoke with him later,
and asked how he liked it in America.
He replied, very nice, however you have confiscated all my works and I am getting nothing at all from them.
Truman got very annoyed at this and so he wrote to the Custodian in Washington that this must be sorted out,
and it was indeed sorted out.
After that I wrote a letter to the Manchester Guardian where I raised the issue
that very many others were in the same position, including myself.
Subsequently, I received a letter from the Custodian in which he wrote once more
that the matter would certainly be dealt with – “in due time".
Then again nothing happened.
And then a scientific attaché from the American embassy in London came to me one day
and said that the Navy, the American Navy, would like
- they are generous with funding and support scientific institutes,
they would like to bring out a book on optics and have produced a great plan and recruited many people.
Then they happened to hear that I was planning such a book.
And if this, my book, matched their intentions, they would give up their plan, which I found very honourable.
And at that time I really was prepared to produce a new book.
To republish my old book was no longer possible and so I wanted to produce a new one,
I already had done some recruitment and presented him with my plans.
These were accepted and then matters went smoothly.
But then the Navy got involved, which is apparently one of the most powerful institutions in the USA,
and then I needed only five more years until I received compensation and the rights to my book once more.
So that was good, now I could have the old book republished, but it is of course quite out of date.
In the meantime, I had started on a new book with the encouragement of many English colleagues
and that succeeded since I had found in Doctor Emil Wolf, a young émigré from the Czech Republic,
a colleague with a good education in optics in Cambridge.
I have now worked with him for eight years on this book which should now be appearing.
I can only give you an impression of all the things that happened in the meantime.
Habent sua fata libelli is then really true, not only when they have appeared, but even before that.
For example, one thing was: One colleague, in the end we had about seven colleagues,
but many more before that who gave it a try, but that did not always work.
For example, one of them delivered a manuscript which we had to refuse completely.
And then he wrote furious letters to the publisher and still demanded his fee.
The publisher did not agree and the case was headed for the courts until we heard
that this was unnecessary, as the man had been imprisoned for deception.
That had nothing to do with us but it freed us from this burden.
Such things always happen.
It was the first time that I had directed such a team, as one has to these days.
I myself did not make much of a contribution to the book.
I have read every word, all corrections, I know it quite well and it was written by Dr. Wolf.
The employees supplied parts of it, but still he rewrote most of it.
He also translated my old style into the English language so well that I can read it as though I had written it myself.
So the book should now be appearing, I was hoping to present you with a finished copy.
I do have one here but it is not quite finished.
It is bound and looks the way it should, but as you know a printers' strike broke out in England a fortnight ago.
Then the only thing that was not ready was the subject index.
This has been defined and is not clear when it will be possible to complete it.
Although I believe that Pergamon Press are doing their very best.
And they have bound this book for me without the index.
Otherwise it is absolutely complete, only the index is missing.
As you can see, it gives rather a voluminous impression.
Now, you don’t need to worry that I am going to read out large parts of this book here,
what I intend to do is to pick out a few small sections.
But first I would like to mention that this book here, although it is rather large,
is only a minimal part of what a real physicist calls optics.
In the old book we already left out the optics of very fast movements, of fast moving bodies,
since that now belongs to the theory of relativity.
In addition, we left out the optics of the creation, destruction and scattering of light.
Since that belongs to quantum theory.
But we have included something which used to be called molecular optics
– which means taking into account the motion of the light-emitting and absorbing atoms and molecules.
Meaning the temperature dependence of the optical effects.
This chapter makes up about half of the old book.
It is also missing from the new book.
Thus everything is left out from the new one which really is of interest for the physicist.
It is therefore really more of a technical book.
And still I believe that it has a certain interest in terms of physics,
because these fine wave phenomena which are involved here appear everywhere else in physics,
in the wave theory of the electron and in other areas and so the interest is not purely optical.
But all the same, my colleagues were all real optical physicists.
I myself am not, and so I was directing a team of people who were of quite a different nature from me.
I myself learned a lot from that.
But now I have said enough for the introduction and would like to pick out a few things.
One section for example ... First I would like to say
that what really appears in the book is the propagation of light seen
as a special case of the solutions of Maxwell's equations for continuous extended matter,
without taking into account the atomic structure or at least with just a very superficial treatment of this,
and the application of this to optical instruments, about which I really know a lot less than about the things mentioned earlier.
All of that was therefore essentially done by my colleagues.
But I learned a lot from this and I believe I can fish out a few points
where I can tell you what this particular effect is actually based on.
Now here among us we have the great optical physicist, Professor Zernike,
and Professor Zernike criticised my first book in the kindest but strict manner, for which I am still very grateful.
And now my main interest is to see Herr Zernike convinced that at least the errors
that were in the old book are not in the new one.
The first thing that I would like go into very briefly, is the theory of thin layers.
And this for the simple reason that it is very important in practice.
One can, for example, apply thin layers on a glass or other transparent surface by vapour deposition or another method,
to change the reflectivity of this surface, to increase it or decrease it as one wants.
The whole problem, seen mathematically, is then this: One has a series of substances which are layered on top of one another.
Each layer has a particular optical refractive index and one wonders how much light exits from there
and how much is reflected back?
How much light goes through and how much is thrown back?
The methods which are used for this today are largely due to a French researcher, Monsieur Abelès.
And he provided us with a very good outline of the matter, after we had worked out these others.
But then a new approach applied known as matrix calculation.
I have a board here on which I would like to draw it.
So here are these layers, these red lines.
And now a beam of light enters here and then we distinguish two cases:
one case where the electric vector oscillates parallel to the incident plane,
that is in wave normal, and the other case where the magnetic vector oscillates like that.
These two … I only want to consider one of these cases.
In any case, each of these two cases is quite independent of the other
and behaves in such a way that one can understand it like this: if this is the direction of propagation
which I want to call Z, and here, let us say that the electric vector oscillates,
and then here the magnetic vector in the Y direction.
It is a system of perpendiculars.
Now, I observe the light, which traverses a layer here, before it enters.
And examine what happens to the light when it reaches the next layer, before it enters once more.
This metamorphosis then includes firstly the transit through the surface layer and secondly the propagation in the next medium.
There are of course simple transformation formulae for each such process,
which...,let us say, for one component here, the electric component U and the other V –
that is the magnetic one here – are transformed 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 electric case, in these two cases.
When one has that, then dealing with many layers, one after the other, is more of the same.
One constructs this matrix, that means, I assume, today every physicist knows from quantum mechanics what a matrix is.
The system of these four figures and the familiar multiplication rules.
Let us 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 if we now want to know what happens in the end.
Then we first have to multiply the matrices together in accordance with the rules for matrices.
This nice recipe was performed there, by Herr Abelès and also in America by Mr.
Billings, and is very powerful.
I would just like to show you quite a simple example that really complicated things can some out there in the end.
The first image deals with the case where we only have three media.
Let us say a glassy substance which is coated with a layer of another material, a thin layer, and above that is air or vacuum.
So vacuum layer and glass.
The thickness of the layer in wavelengths is shown here.
Here is a quarter wavelength, half a wavelength, three-quarters etc.
From here, another scale is used.
That is why it extends so much.
The scale runs up to here – that is one unit – and here are four-tenths of a unit just from here to there.
So that is a very random unit.
The first medium, air, has a refractive index of 1, and the third on the other side, glass, let’s say 1.5.
And the one in the middle, the medium in between, can have any refractive index at all.
And now the result looks like this: If the intermediate medium has a refractive index
which is much greater than 1, then this upper curve applies.
And you can see that one can vary this.
From the reflection here we assume a reflection index – ability to reflect, how much light is reflected.
From 0.04 – that is very low – to a very high value: 0.5 approximately.
And that is repeated periodically with increasing thickness.
But if the thin deposited medium has a refractive index lower than that of the greater of the earlier two,
that is the glass, then it is the opposite, then it goes down, as this curve here shows,
but only very slightly, because this scale is enormously enlarged from here downwards.
And I wanted to show you that as an example of how one can very simply,
if one can place a calculating demon there to work through all such examples.
And that is also how it really happens in industry, where a lot happens.
Now I would like to switch straight to an interesting problem.
By this I mean geometrical optics.
That is the limiting case where one regards the waves as so small that one can speak of rays rather than waves.
There is a method for dealing with this which originated with William Rowan Hamilton.
But then developed later, under the name Eikonal, by Bruns in Germany,
and which I covered in my first book in a form which it acquired from the astronomer Schwarzschild.
This aroused a lot of anger among the real optical physicists.
It is actually very high mathematics and they don’t have much of a feeling for this and attacked me,
only the simple was relevant and this was quite superfluous, and two parties were formed.
Hardly anyone followed them.
When we were planning this new book we were quite open and said maybe we will do it differently this time.
We thoroughly tested everything with all the optical physicists available – the war was on – so we only had the English, American.
And as it turned out we convinced them all, our method is the best, that of Schwarzschild.
So we all converted to using this, what is known as Schwarzschild’s Eikonal,
which essentially consists in regarding the wave no longer as an actually vibrating wave,
all the same the surfaces of the same phase are treated as waves in their propagation,
and the equation which determines this is called the Eikonal equation because that is what Bruns called it.
Now I would first like to tell you, if a representation is perfect, then a light point,
we have the light source Q here, it will, thanks to the imaging system,
produce another light point on the other side of the screen. Point to point.
That is what is known as the absolute Gaussian transformation, which only exists in the roughest approximation.
Geometrical optics, alone, even without the waves, already yields errors which arise from the fact
that the surfaces can never actually be so constructed that this recombination really takes place.
The theory of these errors was first completely worked out by a man named Seidel,
and so the lowest level of errors is called Seidel errors.
There are five of them. Why does it happen to be five?
That is a general problem which I think every physicist should know. Why are there five errors?
These errors are called spherical aberration, astigmatism, field curvature, image distortion and coma.
I would now like to tell you where it comes from, that these are five.
It comes from: If you look at the entry pupil of the instrument here, in other words the opening where the light enters,
and then look at the image plane here, then you have a coordinate system in each
and I can, of course, arbitrarily place the point of entry of the light on an axis, for example on the y axis.
And I name this distance here Y0.
But here I cannot do that, here it lies somewhere or other and has an angle,
which I have called theta, and a distance rho. So there are three variables.
The distance Y0 of the image point from the centre, then the distance – no, of the light point from the centre –
the distance of the image point from the centre in the image plane rho
and the angle that this is deflected, this ray, which comes there.
These must now be combined with quadratic combination, because it is easy to understand
that the figures naturally do not change if I change the sign of all the quantities, so it has to be quadratic.
Thus one can construct three invariant quantities, namely quantities which do not change
if I simply rotate the whole instrument about its axis, then nothing should change at all.
Thus the invariants are essentially Y0^2, Rho^2 und Y0-Rho-cos(Theta).
That is demonstrated in the most elementary mathematics, they are the three invariants of two distances.
But from three quantities one can construct three second order combinations.
Namely the three squares and the three products.
The first combination is Y0 and to the fourth, squaring it again.
And that is assumed.
So it has absolutely no clear meaning.
5 are left and they are the 5 image defects, which are essentially Rho^4 and its square, 0^2, Rho^2, Cosine^2 Theta, there are 5.
The three products and two of the squares.
But now I, and that is one of the few points
where I contributed something positive to the book, have found a method to show clearly what that means.
Which is: I imagine the ray arriving here, passing through the instrument and recombining here.
So that is a concentric sphere which converges on that point.
And the sphere is not exactly a sphere, it is a little distorted.
The distortion of the sphere around a ray, which is now displayed in the figure.
And there you can see 5 possible distortions and beneath each one stands what it is.
So here you have the spherical aberration, proportional to Rho^4, that is 2%, these fourth powers are squares of Rho^2.
That is a kind of flat, plate-like ellipse.
That is the coma. This increases with Rho * Rho^2, Rho^2 * Rho cos(Theta), and also with Rho^3 cos(Theta).
That is a bit of curved surface, not symmetrical. This is astigmatism.
This increases with cos(Theta) and Rho^2.
This is the field curvature, it increases with Rho^2, so it is a paraboloid.
And finally the distortion which increases with Rho cos(Theta), making it a section through an ovaloid.
I say that these figures are new, they were absolutely not to be found in any other book
and they provide a very clear picture of what distortion really is in an instrument which is not completely corrected.
I will come back to these image defects later on.
Right now I would like to switch to a completely different problem:
Is it not possible to make these image defects themselves visible?
And today certain interferometers serve this purpose.
That is instruments which allow a light beam to be divided in 2 and later combined again,
after both parts have travelled different paths.
So that path differences appear.
The interferometer which I use here is a modification by Twyman and Green of Michelson’s famous instrument.
I will quickly run over the principle, as with Michelson one has a glass plate for the division,
which reflects a part of the light that hits it and transmits another part.
Here is a collimator and here is one, and here a flat plate, a mirror.
The light arrives here at the plate, is thrown out here, is reflected here, passes through the plate
and is recombined here by this collimator.
Here, however, in Michelson’s apparatus there is also a plate, a flat plate.
Instead of this flat plate one now uses a very precise spherically ground plate
and places a very well corrected lens in front of it.
What happens then? No, one places the lens in front that one wants to correct.
That’s how it is. Not yet corrected.
The light that arrives here, if the lens were corrected, would then appear,
as it were, to emanate from the centre of this spherical mirror.
But if the lens has defects then they overlay each other, so to speak, over this spherical surface as little disturbances.
And one can photograph that, since when the light is combined with the other undistorted light,
which comes from here, interference occurs which is an exact representation of this phenomenon.
There, for example, you can see the observed, above, calculated below,
the first picture on the left is the spherical aberration, the second is the coma and the third the astigmatism.
I think the agreement between the calculated and the observed interference behaviour is really rather nice.
It is a matter of tiny effects. Naturally I cannot accept this applause.
It was done by someone else, I have also written it down here, but I don’t want to bother them by name.
And now, to stay with interference effects, I would like to show you another nice picture.
It comes from, and here I want to mention the name, from Professor Polanski
who has done a lot of work with diamonds and other crystals and is interested in the nature of their surface structure.
And for this he uses multiple interference, which means that he takes the surface of the crystal
and places on it a very well ground flat plate.
And then he exposes it to light and arranges it so that the beam of light is reflected to and fro many times before it leaves.
And as one knows from fundamental optics, the result of this is that the interference pattern is extremely fine.
The more interference, the finer the pattern becomes.
Now I don’t want to tell you anything more about the detailed theory of the matter,
I just want to show you one of his nicest pictures.
That is the split surface of a mica crystal, which was imaged in the green and in the yellow mercury line.
I have the angstroms here, but it is not of interest.
What you can see from this is that an irregularity runs through the crystal surface here,
quite clearly, a crystalline inclusion or something of the sort.
The order of magnitude which can be resolved in this way, that is very close to the resolution of X-ray images.
I don’t want to go into this any deeper, but just show you the sorts of things that are possible today.
Now I am approaching very insecure ground:
That is, I want to talk about Herr Zernike's famous method of phase contrast, and he is here.
Herr Zernike himself gave us a lecture on that here three years ago,
and I must admit that I did not understand much of it at the time.
In the meantime, under the pressure of the book, I have studied it somewhat
and I now believe that I understand it enough to be able to explain more or less
what it is about to someone who claims, someone who has absolutely no idea about optics.
Two things must be distinguished.
First of all, a fairly trivial mathematical matter.
When, namely, an object is here and a microscope, and the rays come through,
then the object will have an effect on the rays which can be represented by saying
that the amplitude of the light is multiplied by a factor f(x).
Where O(x) here is this direction, some direction in the object.
This function f(x) is in general complex, which means that it signifies a distortion in the intensity and the phase.
The form, let us say: small f x e^i Phi. That is the amplitude and the phase.
Instead of that I will simply write: f x 1 + iPhi + higher terms which I will develop.
The phases must be greater and lower.
Now, if I make a very small cut under the microscope that produces no significant change in the intensity.
Only with transparent substances, in particular, with light that is f1.
And if I then calculate the intensity from it, that means squaring this ...,
or multiplying it with its conjugate, in absolute terms squaring, so it is practically 1.
One can see that here immediately, but also 1 here.
1 + Phi^2, Phi^2 second order, I will leave that out.
So one does not see that at all. What can one do to be able to see something?
That results from a clever thought about the nature of microscopic imaging from Abbe.
According to Abbe, microscopic imaging consists in illuminating the object here.
And I would like to consider the object as if it were a grating.
Then the first effect of the illumination consists in the generation of a diffraction pattern here.
That means first of all a direct ray and then a very much weaker ray of the first order to both sides
and then a second order ray and so on.
They represent the diffraction patterns of first, second and third order from such a grating
and, analogously and generalized, for any arbitrary object.
Then comes the imaging apparatus, it combines all these rays or at least as many of them as it can capture.
If it combines all of them, then a similar image of the grating is produced here,
and if some are excluded, then a dissimilar image is produced.
This is the theory of Abbe in simple, in the simplest outline.
He relates the resolution of a microscope to how many diffraction patterns the instrument can recombine.
Now what Herr Zernike does there is this, he says: If he could turn the 1 in this function into an i,
then it would go like this: i * (1 + Phi).
And if I square that, then it has a linear member.
I^2 is 1, but this gives 1+2Phi+i squared.
Then there is only one i there of the kind that one requires. So what must I do?
I must somehow manage to turn this 1, in this transparency factor, into an i.
But what does the 1 correspond to?
It is very easy to see that it is the direct ray of zeroth order.
And the phi corresponds to a higher order.
Herr Zernike, forgive me if you think this is over-simplified, but in my opinion it may be pedagogically usable.
Then one just needs to interpose something here, a body, a disk,
which changes the phase of the wave in transit so that the 1 becomes i.
Because the i means a phase of pi by two, of a right angle.
And that can be done.
One has a plate which changes the phase, but only that of the light in transit, not of the diffraction patterns.
And then this function is changed into this one, and the effect is there.
It changes the phase effect into an amplitude effect.
And that is the basic idea, which we of course extended, you can look up the precise theory in the book.
I would like to show you two images of such phase figures.
That is a splinter of glass, with A showing the direct image, where you hardly see anything, just the edges of the glass splinter.
B and C are phase contrast images at two different apertures.
Here you can see a lot of detail which you don't see here.
So that is the success of this method of Zernike’s.
The next image shows something organic, in fact frog epithelium.
A and B are a direct image with two apertures, and C and D are a phase contrast image, again with two apertures.
In A and B you can see extraordinarily little.
In C and D you can see a lot of detail here, and here too, here they are exactly reversed in the intensity,
that depends of course on random factors of the apertures and suchlike.
But one can see here the extraordinarily high resolution which one can obtain with this method.
Now I will return to my image defects.
If you had an ideal image, meaning a point of light is recombined into a point of light, the question is,
what does it look like from the point of view of wave theory?
Not just as rays which emanate from a point and recombine here, but as a wave which is emitted here,
then refracted, and is concentrated here as a spherical wave.
And how does such a spherical wave behave in the centre, under the assumption, of course,
that this spherical wave is not closed but, as in every instrument, limited by the aperture.
The original theory was developed as far back as the year 1885 by the German Lommel, and it is still the best.
Although Debye, in 1909, took another approach with integrals,
which added a few new thoughts, still Lommel's calculations are quite outstanding and must be used.
So we used Lommel’s formulae to work through the matter, or rather we took it from a work by Wolf and Linfoot,
and I would just like to show how comfortable it looks in the vicinity of a focus.
That is where one thinks that light converges nicely to a point.
But what really happens is much more chaotic than that.
So here is the ideal focus. The numbers written in there represent intensities.
That is for my eyes very hard to read. Here it says 217, 477 … 677.
Then there are a lot of little figures all around.
So you can see that there a few high maxima next to the focus,
which is, of course, also a high maximum, and then smaller maxima here and a lot of other maxima around.
That is how the image looks in reality.
It is quite a complicated mountain range.
If one sections it like this, in the focal plane, then one obtains an image which one can examine here.
In the centre, a high mountain accompanied by ever decreasing smaller mountains.
That is the usual diffraction pattern. The Airy pattern.
A maximum and then the little tails at the sides.
But here it is three-dimensional and what we have is only one of many sections.
A few of them also appear in the book.
So that would be the ideal Gaussian transformation.
But what happens now … no, unfortunately my diagrams are in the wrong order.
Before I mention what happens, if I now consider the image defects, the geometrical image defects,
I would like to touch on another point.
One of our first colleagues, but who left us later, Professor Gabor of Imperial College in South Kensington,
he published what is known as a reconstruction method, which is based on the following.
Here I have to return to this figure of Abbe’s.
It plays a role in electron optics.
In electron optics one has problems with the resolution of the instrument
which is limited by the …oh, I don’t want to go into the technical details, I just want to say what it is about here.
You will remember: The object emits its diffraction pattern, the diffraction pattern is recombined by a lens,
and if it combines well – everything that is there, then a similar image results.
Gabor had the idea of dividing this process in two.
Namely, to replace the lens with a photographic plate here, and first to photograph the diffraction pattern of the object,
then to illuminate the photograph again in the same way and so to reconstruct the image.
When they first told me this, that is nonsense, that won't work.
Because the diffraction pattern does not contain the intensities alone, but also the phases.
But the photographic plate does not take any notice of the phase, just the intensity.
So what one now obtains is a distorted image because the phases are wrong.
The great trick which Gabor achieved is by observing with very narrow bundles,
and electron microscopy does not deal with anything else, to show that one can arrange very narrow bundles in such a way
that the phases simply have a negligible effect.
And that one obtains good images by dividing the whole procedure of microscopy into two parts.
One makes firstly a diffraction pattern, and then one photographs the diffraction pattern and obtains the object.
The same idea, by the way, was also used by Bragg in crystal theory,
where X-rays are used – they provide only the diffraction pattern.
And there he thought: Can’t I use the diffraction pattern to obtain a real image of the crystal by photographing it again?
Of course it does not work there, because there a wide angle is involved.
But still Gabor’s investigation showed what had to be done to carry it out technically.
But it has, I think, as far as I know, never been carried out completely.
I would like to show you one of Gabor's examples. Here we have the original.
There are a few names of optical physicists: Newton, Huygens, Young, Fresnel and Bohr.
Here they photographed the diffraction pattern, which of course has no trace of similarity with the original.
And this diffraction pattern was sent back through the same apparatus and reproduces the original quite legibly,
so I can read it from here.
Once again Newton, Huygens, Young, Fresnel and Bohr.
Of course it includes small distortions, because the phases cannot of course be avoided altogether.
I wanted to insert this here.
That is also included in the book.
And now I finally return to the diffraction theory of image defects as the last point.
These five image defects which I named, and which I keep forgetting, much to your amusement,
they were calculated with ray optics where one assumes that light consists of real rays which are refracted.
Now what becomes of this if they are waves?
Then one must take these rays and regard them as carriers of waves and also take into account the phases.
Of course that leads to very involved integrals according to Kirchhoff’s theory of diffraction,
and again it is Herr Zernike whom we can thank for the correct method.
I developed the first version of this theory in my book, as well as I could manage it;
the whole book was ready within two years.
I had just one month for this work.
There I grabbed the formulae as they were, and tried to evaluate integrals.
This did not produce very much benefit, but still an insight into the matter.
Later it was taken over by various people, by Blaser and by Zernike himself.
And Herr Zernike found the right method.
The right method consists in developing the waves with particular functions,
which are now known as Zernike polynomials, and these are polynomials,
meaning XY multiplied together, provided with factors and added together.
And these polynomials have this property: They are defined in circles.
The circle, the aperture of the instrument … and if such a polynomial is taken in the original X,Y coordinate system
and this coordinate system is rotated into a red X‘, Y‘
– one can of course in principle express X’, Y’ in terms of X, Y and the point of rotation, in the usual way.
X dash cosine, Y dash sine and so on.
And then the new polynomials which are derived from the old ones should be identical with the old, apart from a factor f
which depends only on the angle of rotation.
So the polynomials are thus essentially uniquely defined, which is also a normalisation, and as Herr Zernike showed
they are the right elements on which this theory must be constructed.
To go into the theory itself is of course pointless here.
I would just like to show how one makes photos of instruments, of lens systems,
which still have the various defects, which have not yet been fully corrected.
How the image appears in terms of wave theory.
So I have three images. The first involves a meridian plane, in other words a plane which includes the ray itself.
And this ray is like that. ...or contains the axis of the instrument.
That is the first, the spherical aberration.
The second and the third involve planes at right-angles to the axis of the instrument. Like this.
And the first involves coma and the last involves astigmatism.
These are at the same time generalisations of my second figure
where I showed you these small areas which represent the geometrical distortion,
and also the image just shown of an ideal bundle in the focus.
If you now overlap these two things, the geometrical error with the wave nature, such figures arise where the isophotes,
that is the lines of equal brightness, are drawn; here is 75, goes down, here is 5, 2, 1 here it goes back up to 9.
You can see the images are really very convoluted.
This, as I already said, is an image of the spherical aberration in the meridian plane.
The light propagates in this direction.
This continuous line is what is known as the Gaussian line of geometrical optics, which I won’t go into.
Up here you have three theoretical figures, calculated from the formula, for the coma,
where, as those involved with optics know, this remarkable figure is distorted to one side.
That is connected with the fact that the little area was so asymmetrically distorted.
Up above are the theoretical images for various cases and here below are the corresponding experimental images.
And if one compares this image with the theoretical one can see how nicely it corresponds here
or they are not so exactly aligned, but simply to show how well experiment and theory agree with each other.
The last figure, that shows the same for astigmatism, but I don’t want to go into detail.
In the book can be found the exact theory for diffraction in the few cases where it is possible.
This means for spheres and half-planes.
If you have an aperture consisting of a half-plane and light enters, what happens to it?
That is one of the famous problems that was first solved by Sommerfeld.
But today this is treated quite differently by today's mathematicians.
Not with plurivalent potentials as Sommerfeld did, but directly with what is known as dual integral equations.
But I cannot trouble you with this either. Those are just questions of method.
The figure that I want to show you now just shows how it looks if one really works through these formulas of Sommerfeld’s.
He never did it himself. And here I have the phases in one figure and the intensities in the other.
And it is the case, that light … here is clearly the shadow, here nothing happens.
Here are the lines of equal intensity and here of equal phase.
So you can see here how the shadow is produced.
These are the rays which are deviated a little upwards.
Here is the reflexion, the light which falls here on the shadow against the body,
is deflected upwards and interferes with the incoming light and produces such interference patterns.
Here is the same for the phases. Very complicated figures are produced here.
This just to show how today one can work through Sommerfeld’s formulae in full detail.
The last image that I want to show you involves a theory which was originally developed by the French physicist Brillouin,
who predicted this phenomenon: Imagine a container and a liquid in which very short sound waves are generated,
which can be done today electronically.
Let us imagine these sound waves travelling upwards from below.
Then these acoustic waves are locations where the refractive index changes - periodically changes. So they are a grating.
And if I now let a beam of light pass through it at right angles, it will be diffracted.
That sounds like a pleasant diversion, but it is much more.
It happens to be the best way so far, and perhaps the only way of observing the optical properties of small bodies.
I can still produce such very short acoustic waves even in tiny crystal mirrors which I can hardly see.
In this way I can determine how fast light propagates – from the differences that I see.
And from that I can calculate the elastic coefficient of the crystal.
With this method the elastic constants of rare crystals are today known very precisely.
But here I only want to discuss the theory for liquids and not for crystals.
There were a lot of approaches around, from Wannier and others and then from the Indian, Sir C.V.
Raman, who was also here three years ago.
But who only wanted, or only was able to handle a remote marginal case.
In Edinburgh with me at the time were Noble, a Canadian, and Bhatia, and Indian,
and I brought them together and suggested the method to them and they then worked out the theory.
There it involved this, I just want to sketch it quite briefly.
Here is the container, down here is the vibrating crystal and here the acoustic wave runs upwards.
Whether that is a static wave resulting from being reflected down again,
or a moving one, makes no difference at all.
Sound is so slow compared with light that that … that a moving wave stands still
and then one has a light source here and an imaging instrument, with a parallel light bundle,
it comes in here and is then recombined here by a lens, and the focal plane.
Here the interference takes place, it functions like a grating through which it passes.
The angle of incidence must be very small, or one sees nothing.
The method that we applied here was familiar to me because it was essentially the old perturbation method is
which was introduced into the quantum mechanical theory by Heisenberg and Jordan and me.
With this perturbation method, which is a very curious case of degeneration,
the two succeeded in completing the calculation, and now I would like to show you this image.
Up here is an experimental picture from the Indian Parthasarathy.
And what you may be able to see there most clearly is the asymmetry right and left.
Although it surely looks as though it ought to be quite symmetrical.
If one thinks about it in detail, it is not symmetrical,
because the ray has a very small angle of inclination of 2 or 3° at most.
But this minuscule difference has the effect that, for example,
here only three lines appear above and five here below.
And here the difference is even greater.
And these intensity relationships are, as you can see, very convoluted.
And here the angle of incidence is stated up here.
It is difficult to read: Null, 0.06 etc.
… degrees! And now it says here: F1 and M2 are the numbers of straight and non-straight lines,
one to the right and the other left of the middle of the straight-through ray.
And there you can see the same asymmetry.
Here it is still fairly symmetrical with a really small angle,
here it already starts to become very asymmetrical and then it strangely becomes symmetrical again.
So that is no simple law, where it reverses from asymmetry back into symmetry.
That is well expressed in the theory. Here are the theoretical values in brackets, and the observed without brackets.
[inaudible 43:33]... would say 6:4 here and 5:5 here and 9:2 and 6:2 here, so there is no absolute agreement,
but it is of course also an approximation, what one wants to regard as still visible.
Still, we were quite content when we had that.
In any case we were now able to subsume all marginal cases that occurred in the literature under this theory.
Here you have a short overview over this book.
I would just like to add just one thing, a very important chapter is now this:
Here we have always regarded the light source just as an illuminated point.
But the light sources have a finite size.
This extension has the effect that the phenomenon of partial coherence occurs,
that parts of the light source no longer oscillate independently of other parts, but are coupled together.
There is a very large chapter on that which comes from my colleague Wolf,
and where Zernike’s results are once more exploited on a large scale.
In addition, the book contains some appendices, of which one was written entirely by me.
There is a generalisation of geometrical optics, known to mathematicians as variation calculus,
and I have presented them there to show that the main optical phenomena still take place to a much greater extent
if one takes complicated functions in place of the refractive index as the single characteristic quantity of substances.
That is necessary if one is engaged in electromagnetic optics, that is to say electron microscopy.
This is also covered in an appendix by Dr. Gabor.
Finally, I want to show you how the thing looks today, apart from the index.