The Waviness of Light - 40 years of optics
A Lecture for John Little's Retirement by Richard M Sillitto Delivered in the University of Strathclyde, in 1986 at an event chaired by Richard Thornley First, thank you for allowing me to participate in your tribute to John Little. I've known John since 1946, when he was an undergraduate in Edinburgh, and I returned from war service as a research student. Since then we've kept in touch, and I very much hope that in these 40 years John may have learnt a little from me - I'm acutely conscious of having learnt a lot from him, and I'd like to feel that at least a small fraction of my debt to John has been cancelled out. I'd like to tell you a little story about John which has more to do with my purpose here today than may appear at first sight. One Saturday morning, many many years ago, my wife and I were walking down the North Bridge in Edinburgh. In those days the Bridges was one of Edinburgh's principal shopping streets, and Patrick Thompson's ("P.T's"), along whose frontage we were walking, encapsulated the standards and aspirations of comfortable, middle-class Edinburgh. Ahead of us we were faintly aware of an unusual pattern of movement on the crowded pavement. As we drew closer, we saw, above the crowd, a familiar head with a shock of black hair, apparently moving to and fro across the crowd in simple harmonic motion. Soon we could see that this apparent SHM was in fact the projection, perpendicular to our line of sight, of a uniform circular motion - how tolerant those other pedestrians were! When we were within hailing distance I was just about to speak to John when he, noticing me, pointed - as he gyrated - to a notice in the shop window which read:
Discussing any topic with John, you're liable to find that after an amiable start, an element of cross-purposes seems to appear. John has started to walk round the subject, and is seeing it now from a standpoint different from the one you initially shared. You have to choose whether to stay put and allow the cross-purposes to get crosser, or to follow him round, and share his experience of the diversity of the subject. It's reminiscent of the work of the French Cubist painters early in this century who discovered the trick of displaying in a single picture aspects of the subject which can be seen only from a number of different viewpoints. Once you accept the apparent incongruity of such compositions, you realise that the painter is showing us more about the subject than the static observer can hope to perceive.
There are difficulties about this combination of apparently incompatible aspects. The holographer who tries to reconstruct the real image from a Fresnel hologram with too wide a beam, overlays on the screen different views of the subject, and generates diffuseness; whereas the painter can separate them spatially, and combine them into a harmonious composition. Scott Fitzgerald, the young American writer of the 1920s, wrote that it was a mark of genius "to be able to hold in the mind simultaneously two contrary views, ... and continue to function."
But quantum physics requires us to do just this, and - like the Cubists? - has devised formal procedures for doing it - they comprise what's called "the transformation theory". When I was asked to review the developments in some branch of physics during John's working life, it seemed to me that I could link John's interest in the superficially paradoxical view of nature which quantum theory challenges us with to my abiding interest in optics which has been, in this period, one of the most exciting and most spectacularly developing branches of physics. Of course, this has been an exciting period of progress in many fields. In society we have seen the decline of the safety pin and the rise of the Velcro fastenerIn optics we have seen - I hope and trust - the decline of the photon and a deeper understanding of the wave-picture: there may be room for photons in high-energy physics, but they've been a confounded nuisance, and at times a source of bad temper and destructive cross-purposes, in optics. We've seen the development of coherence theory, and out of it, part way along its progress, an elegant, and powerful, and profoundly satisfying discipline of quantum optics. We've seen the development of the laser, of non-linear optics, of the optical image processor, of phase-conjugate optical devices, any minute now the optical computer; on the High Street, the laser light shows at Christmas, and shops overflowing with binoculars, cameras, zoom lenses and what-not at prices which, in real terms, are trivial compared with those of 1946. But what I want to share my thoughts with you about is not optical hardware, but how the way we think about light has developed over the past 40 years. Much of what I say - in particular the last part, will relate to work by an old friend E. Wolf, whom John I know, remembers as one of his teachers in Edinburgh.
In 1946 there was no quantum theory of light-fields. There was of course QED, but if you pick up Heitler's Quantum Theory of Radiation which was then the standard work in this area, you have to search long and hard to find anything about optical fields or optical phenomena. Of course, we know that "an e-m field is an e-m field is an e-m field", so if there is a quantum theory of radiation doesn't it encompass optical fields? Well, yes, I suppose so; but in the real world these great generalisations need to be shaded, and qualified, and interpreted.
In the years just before 1946, national needs led to physicists and electrical engineers being tossed into blacked-out pots and stirred up together. Among the things they discovered in this process were substantial areas of common ground, and significant areas of mutual incomprehension. Engineers knew that radiation from independent sources of radiation (radio transmitters) can produce interference (it's simple trigonometry, after all, to add two cosine functions together!);What we did have in 1946 was an appreciable amount of theory about optical coherence, and a very powerful theory - the result of intensive wartime work on radar in the UK and USA - of random signals and noise. The merging of these two trains of thought, mainly by E Wolf (in Cambridge and Edinburgh) and by H H Hopkins (in Imperial College, London) produced by the mid-1950s a very highly developed theory of physical optics, one aspect of which was to prove particularly significant in the next 15 years. But before I talk about that, let me go back a bit.
First of all what is coherence of light? If an extended source is radiating light, then in general every surface element of the source is radiating independently of every other. When the waves from the different bits of the source are added together to give the resultant field at some point, the resultant is a superposition of components with randomly different amplitudes, phases, and even - within limits - different frequencies. The result of this superposition of random contributions at some point P1 is unlikely to be the same as at some other point P2. What coherence theory aims to do it is to predict how similar or how different the resultant fields are at any two points P1 and P2. The extent of the area over which the fields have an appreciable resemblance to one another is called ‘the area of coherence’. To produce clear interference or diffraction effects we have to be working with light from points within a coherence area.
The ansatz encountered in so many undergraduate text books - "the light field can be described by E=E0cos(wt-kz)" - is clearly an unsatisfactory starting point for the treatment of real fields from real sources; but we usually reassure ourselves with the thought that we can surely superpose ensembles of such primitive representations by means of a Fourier integral. But can we, really? The earliest source I've encountered for a clear and purposeful statement that it is not satisfactory to construct a theory of optical processes in terms of elementary components whose amplitudes, wavelengths and phases are all unobservable was a paper published in Madagascar in (I think) 1934; the author's aim was to develop a description of light-fields in which the unobservable oscillations did not figure. Whether Wolf knew of this paper I don't know, but by the mid 1950s he was able to claim that the theory of partial coherence ... operates with quantities (namely with correlation functions and time-averaged intensities) that may in principle be obtained from experiment. This is in contrast with the elementary optical wave theory, where the basic quantity is not measurable because of the very great rapidity of optical vibrations.And from a different standpoint, which is not that of a book on classical physical optics such as Born and Wolf wrote, one can add that quantum theory tells us that we could not know the phase of the light field of ‘the elementary optical wave theory’ if we knew its intensity, whereas the observable functions of coherence theory don't run foul of quantum theoretic inhibitions.
The coherence functions are statistical functions which specify the degree of correlation - or similarity - between the light fields at different points, or different times, or both. And there are four of them.
They are all formed by multiplying together one complex function - describing the field at one point/time - and the complex conjugate of another - describing the field at another point/time. If one contains the (unobservable) frequency w, and the other does too, then in one case this frequency appears in e-iwt, and in the other (complex conjugate) the frequency appears in e+iwt. When you multiply these together, the frequency disappears. If at P1 at a particular time the frequency of the field oscillation were w, and at the other point it had a slightly different value w' then in the multiplication we would get exp{-iwt} x exp{+iw't} = exp{i(w-w')t} which only contains the difference between the two frequencies. If the light has a narrow spectrum then w-w' is very much less than omega; typically w and w' might be about 1015 per second, and w-w' is approximately equal to 109 per second; variations at that rate can be detected photo-electrically - I think this was probably first shown in the 1930s in Glasgow by John Thomson - and coherence functions described only these relatively slow and observable variations. The measurable functions of the classical coherence theory are then the mutual coherence mutual intensity auto-correlation and the intensity. It turns out that if you know the mutual intensity across the whole of any surface through which the light passes, you can predict it across any other surface through which the light passes; and from this it follows that if you know the mutual intensity across any surface you can calculate the intensity every where else. The coherence functions can be pictured as propagating through space, and the propagation can be described either by a differential wave-equation (in six or seven variables), or by an integral which sums - on the second surface - the secondary waves (shades of Huygens and Fresnel!) from the first surface. And if the first surface is positioned right up against the source, then you have the solution of Verdet's problem of calculating the coherence of the radiated light from any specified source. And these propagation laws are the fundamental propagation laws of light-fields. In the mid-1950s two very remarkable experiments were reported - one by R. Hanbury Brown and R. Q. Twiss, working at Jodrell Bank; and the other by A. T. Forrester, Gudmundsen and Johnson, in USA. Both could be interpreted, up to a point, in terms of coherence; but since the use of fast photo-electric detectors was crucial to both experiments, it was felt that "a fuller analysis ... must take into account the quantum nature of the photo-electric process." At this point people who felt that the quantum nature of the photo-electric effect could be taken into account by talking about 'photons' got themselves into terrible muddles - and, very often, bad tempers.
The Hanbury-Brown and Twiss experiment
The Forrester-Gudmundsen-Johnson experiment
What is measured in this experiment is the correlation between the intensities at two different points in the field produced by a small source. It turns out that this correlation can be expressed in terms of the mutual intensity at these two points, and the existence of this correlation is perfectly intelligible in wave terms (R H B and R Q T had previously performed the experiment at radio frequencies, and it is done with sound waves in some university U/G laboratory classes!). But if you do the experiment with photo-multipliers and look for coincidences between the detection of 'photons' at the two cathodes you have great difficulty explaining why 'photons' do or don't 'arrive' simultaneously at the two detectors - particularly if you have a beam-splitting mirror in the system. Indeed, the distinguished Hungarian physicist L. Janossy suggested that the effect probably doesn't exist, and that if it does "this will require fundamental modifications in quantum theory".
It was possible to create difficulties in understanding this one if you said : the 'photons' with frequencies w0+D and w0-D must have been emitted by different atoms, so how could they interfere? About both experiments, some people said the reported results were obviously impossible. (Recall that at the British Association in 1876 Kennelly and Heaviside defended themselves from attacks in the words : "they said our statements were impossible, to which we made reply that we had not said that they were possible, but only that they were true"! ) It's a simple-minded view of a 'photon' which suggests that it contains only one frequency; and it's a ludicrously naive view of a light field which regards it as a hailstorm of photons. The only way to construct a photon-picture of these experiments is to look for some theoretical picture which does explain them, and then design 'photons' which contain the essential features of the satisfactory explanation. What is it that we (but count me out!) want the photons for? Simply to embody the notion that matter exchanges energy with radiation fields in discrete amounts. But the propagation of the field - and of its coherence properties - is a wave process, and the wave-features must figure prominently in any quantum theory of optical phenomena. In 1963, then, it seemed entirely reasonable that a quantum theory of optical processes - light-fields, their propagation, and their interactions with sources and detectors - should appear under the title : On the quantum theory of optical coherence. In essence what R. J. Glauber did in two epic papers published in March and September of 1963 was to show that if you replaced the variables in the coherence theory by suitable operators, then the various coherence functions could be associated with quantum operators and their eigenvalues and eigenfunctions.
When you do that, something very remarkable emerges from the theory. Usually in quantum mechanics we work with operators which have real eigenvalues, and we say that the real eigenvalues constitute a catalogue of the results you could get in measurements of the corresponding variables, and then there are rules for calculating the relative probabilities of these various possible results. Now, in quantum optics the electric field in a light-beam is represented by two sets of operators. One of these sets is associated with the possibility of the radiation field's increasing its intensity by
Quantum optical theory developed dramatically. Willis Lamb, the eponym of the ‘Lamb shift’, was one of those who in the 1950s demonstrated by impeccable theoretical arguments that the ‘optical maser’ (or laser) wouldn't work. When it did, Lamb was so mortified that he devoted most of his efforts over the next 10 years to working out, in detail, why it does work, and how it works. It's "a difficult problem in the non-equilibrium statistical mechanics of an open system" - but thanks to Lamb we probably have a better theory of the laser than we did of the thermionic valve triode oscillator!
Because quantum optics, unlike classical physical optics, deals with the interaction of light with its sources and detectors, as well as its propagation, it can treat the non-linear interactions of light with matter, so we have a theory, of non-linear optics. The vibrational energy of the material the light is passing through can be included in the theory in a way very similar to the e-m energy of the light field so the theory could be extended to treat, e.g., parametric processes and Raman lasers. And from the work on non-linear optics in the 1970s has come the excitement of optical bistability, and the prospect of the optical computer.
Meanwhile, back at the ranch, what was being done about optical coherence? First, ideas from coherence theory and quantum statistical thermodynamics were being injected into the somewhat arcane field of radiometry, which concerns itself with the transport of energy in light-fields. This is an important area, interesting results are coming from it.
And then, in the mid-1970s Wolf, Mandel, Carter, and one or two others initiated a significant development in the study of coherence. Hitherto it had been mainly - and most usefully - a theory of the correlations which developed in narrow-band (or ‘quasi-monochromatic’) light. In the 1970s it has developed into an effective theory of polychromatic light fields. What had to be done to achieve this sounds simple if you describe it physically, but there were some quite subtle mathematical difficulties - so subtle that only people with very good mathematical backgrounds would have known that they were there (there are some things it's better not to know about!). In effect, what was done was to express the coherence functions, not as functions of the fields at P1-at-time-t1 and P2-at-time t2, but as functions of the spectral components of frequency w at P1 and at P2. This doesn't sound like a big step to have taken, but it has made possible substantial advances in radiometry; and very recently indeed it has produced some very remarkable results about spectra. Questions about spectra weren't easy to ask as long as the frequency didn't appear in descriptions of coherence. You may think that I've lost my way here - didn't I say earlier that one of the vital steps in coherence was the elimination of the unobservable high-frequency oscillations? Yes. But we're not now putting them back in - the coherence functions of this new branch of the subject do have the high-frequency oscillations eliminated, but we can still assess the contribution to the coherence or incoherence of a light-beam arising from a particular narrow region of an inherently broad spectrum. It's simply that we can put a narrow band filter in front of the source, or the detector, when we measure the degree of coherence, or we can imagine that to have been done when we do calculations on paper. So what can possibly come out of that, that's new? The most striking result so far is that the spectrum of a light-beam can change as the light travels in free space. Most people find that surprising. But in fact, experiments have been known for a long time now - I did one as an undergraduate which could have been interpreted in these terms, but wasn't.
Let me describe this experiment: to be added This perhaps seems a pretty trivial experiment - or at any rate, it's one whose explanation doesn't require any recondite ideas. Fine. That's what I want you to believe - as light propagates its spectrum may change, for very simple reasons expressible in terms of the superposition of waves. In March of this year Emil Wolf published a short paper in Physics Letters (Mar 31, 1986) which showed that (i) in general, the spectrum of light can be expected to change as it propagates, so that the spectrum of the light-beam in the far-field region will not in general be the same as that of the source;Although the effects observed in that 2-slit experiment are easy to understand, the statements I have just made seem rather more portentous. If they lead to no stranger effects than those in my second year U/G experience, we'd be inclined to agree that maybe coherence theory is a stuffy and pretentious way of describing things that are physically rather obvious. But wait ...
Whether or not Wolf's ideas are ultimately judged relevant to the quasar red-shift problem, the effect of spectral non-invariance certainly exists : in a paper soon to be published two experimenters at Rochester have demonstrated it with a circular white-light source and a propagation distance of one metre. The spectral shift in this case agrees very well with the theory when the appropriate experimental parameters are inserted - and for this case it’s a blue-shift!
In work which hasn't yet been published - and my talking about it does not constitute publication in any quotable sense, Wolf has shown the following thing: Suppose you have an extended 3-D source, every region of which is radiating a spectral line whose profile is Gaussian, and if the correlation between the radiated components from P1 and P2 is a Gaussian function of P1 - P2 only, then in the far-field zone the spectrum is still Gaussian but its peak is shifted, compared with that measured at the source, and the line-width that is slightly reduced. The shift may be very large - appreciably greater than the line-width - even when the spatial extent of the mutual coherence in the source is only one or two wavelengths; and the shift is a red-shift.Now suppose some astronomical light source has these coherence properties, and an astronomer, looking at its light in a spectrograph, assumes the shifts to be cosmic red-shifts, he will work out what the velocity of recession of this body must be, he will convert that velocity into a distance by means of Hubble's Law, and will conclude (for the illustrated shifts) that the source is far outside our galaxy. But this shift has nothing to do with relative motion - it applies for sources which are stationary relative to the observer; it is for most practical purposes independent of distance, and it results simply from the superposition, at the detector, of light from a partially-coherent source. Some astronomers are puzzled by the fact that many quasars are so bright that we should expect them to be quite close to us - certainly within our own galaxy - yet they exhibit red-shifts which, conventionally interpreted, seemed to place them a long way out towards the edge of the observable universe. Some have tried to devise ‘non cosmological’ explanations, i.e., explanations which don't rely on Doppler frequency shifts in an expanding universe. Wolf has certainly found such a mechanism. At first, when I read this work, I thought it must fail to account for the quasar red-shifts because in the Wolf theory lines with different widths exhibit different red-shifts : and then I found some quasars show different red-shifts for different groups of spectral lines. It's hard to see how these effects could have been predicted without the recent developments in the study of coherence, and they hold out exciting prospects for future work. Someone or other - I don't have the reference available - wrote that: "The world is so full of such numbers of thingsI rather think John Little has always believed that : I hope he always will. © Richard M Sillitto 1990
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