Beamsplitting Interference phenomena are certainly among the most exciting phenomena in the whole of physics. In the following we will concentrate mainly on interference of weak fields; i.e. the beams contain, on average, only a few photons. The principle of classical interference is as follows: a light beam is split by an optical element, for example by a semitransparent mirror or a screen with several very small apertures, into two or more partial beams. These beams will take different paths and are then reunited and form interference patterns. The first step, the splitting of the beam into partial beams, plays a decisive role; light beams coming from different sources (or from different spatial areas of the same source) do not interfere with each other! We start our discussion of interference with an analysis of the action of a beamsplitter. To form a realistic idea of this device, let us imagine a semitransparent mirror. (Our considerations apply equally well to a screen with two apertures. We could also generalize to cases of unbalanced mirrors, with reflectivity different from 1/2, or screens with apertures of different size.) The classical wave picture can describe interference phenomena without any great effort: the incoming beam is split into the reflected and the transmitted partial wave, and each of these waves contains half of the energy. The process of splitting becomes conceptually difficult only when we think of the beam as consisting of spatially localized energy packets, or photons.
Beamsplitting Interference phenomena are certainly among the most exciting phenomena in the whole of physics. In the following we will concentrate mainly on interference of weak fields; i.e. the beams contain, on average, only a few photons. The principle of classical interference is as follows: a light beam is split by an optical element, for example by a semitransparent mirror or a screen with several very small apertures, into two or more partial beams. These beams will take different paths and are then reunited and form interference patterns. The first step, the splitting of the beam into partial beams, plays a decisive role; light beams coming from different sources (or from different spatial areas of the same source) do not interfere with each other! We start our discussion of interference with an analysis of the action of a beamsplitter. To form a realistic idea of this device, let us imagine a semitransparent mirror. (Our considerations apply equally well to a screen with two apertures. We could also generalize to cases of unbalanced mirrors, with reflectivity different from 1/2, or screens with apertures of different size.) The classical wave picture can describe interference phenomena without any great effort: the incoming beam is split into the reflected and the transmitted partial wave, and each of these waves contains half of the energy. The process of splitting becomes conceptually difficult only when we think of the beam as consisting of spatially localized energy packets, or photons.
Interference by division of wavefront. The two-slit experiment Preliminary remarks In this chapter we shall be concerned with various types of interference effects exhibited by matter waves. It might be argued that all quantum mechanical phenomena are ultimately a consequence of interference, being governed as they are by a linear wave equation. Of particular interest here is the analogue for material waves of Young's double-slit experiment in optics, a classic problem in the history of quantum theory. Feynman, Leighton & Sands (1965) state that it ‘ … has in it the heart of quantum mechanics. In reality, it contains the only mystery’. We shall also treat the process of tunnelling into classically forbidden regions, a phenomenon which also involves interference in an essential way. To observe the interference of waves issuing from a common source we must split the emerging beam into two spatially separated coherent parts and subsequently recombine them (the superposition of waves from physically independent sources is a many-body problem – see §§8.8 and 12.7). As in optics there are two methods of achieving this: by division of wavefront where, for example, a wave falls on a screen containing two apertures, and by division of amplitude in which a wave is incident on a semi-transparent barrier. Interference effects achieved by the first method, discussed in this section, have been observed for electrons (Jönsson, 1961; Lichte, 1986; Tonomura et al ., 1989), neutrons (Zeilinger et al ., 1988) and (helium) atoms (Carnal & Mlynek, 1991).
How can we construct a picture of the photon from the wealth of observation material available to us? The photon appears to have a split personality: it is neither a wave nor a particle but something else which, depending on the experimental situation, exhibits a wave- or a particle-like behavior. In other words, in the photon (as in material particles such as the electron) the particle–wave dualism becomes manifest. Whereas classically the wave and the particle pictures are separate, quantum mechanics accomplishes a formal synthesis through a unified mathematical treatment. Let us look first at the wave aspect familiar from classical electrodynamics, which seems to be the most natural description. It makes all the different interference phenomena understandable, such as the “interference of the photon with itself” on the one hand and the appearance of spatial and temporal intensity correlations in a thermal radiation field on the other (which are obviously brought about by superposition of elementary waves emitted independently from different atoms). It might come as a surprise (at least for those having quantum mechanical preconceptions) that the classical theory is valid down to arbitrarily small intensities: the visibility of the interference pattern does not deteriorate even for very small intensities – the zero point fluctuations of the electromagnetic field advocated by quantum mechanics do not have a disturbing effect – and is valid not only for conventional interference experiments but also for interference between independently generated light beams (in the form of laser light).
Beam splitter Compared with other fundamental experiments in physics, optical tests of quantum mechanics are often distinguished by their simplicity. Most quantum optical experiments do not require a whole industry - an optical table of equipment and a few people are often sufficient. Good ideas, good research problems are more important. “Research is to see what everybody has seen and to think what nobody has thought” (Jammer, 1989). A simple optical beam splitter, for instance, is already a nice device to demonstrate the quantum nature of light. Quite a number of puzzling quantum effects have been seen by splitting or recombining photons at a small cube of glass. Additionally, the beam splitter serves as a theoretical paradigm for other linear optical devices. Interferometers, semitransparent mirrors, dielectric interfaces, wave guide couplers, and polarizers are all described sufficiently well by a simple beam splitter model. The quantum effects of almost all passive optical devices can be understood assuming appropriate beam splitter models. (It's all done with mirrors.) Heisenberg picture An ideal beam splitter is a reversible, lossless device in which two incident beams may interfere to produce two outgoing beams. For instance, a dielectric interface inside a cube or plate of glass splits a light beam into two, see Fig. 5.1. We may reverse this situation by sending the two beams back to the cube, where they interfere constructively to restore the original beam.
The previous chapter surveyed part of the rich variety of physical phenomena that can be understood using the ideas of quantum physics. Now that we are beginning the task of looking more deeply into the subject we shall find it useful to concentrate on examples that are comparatively simple to understand but which still illustrate the fundamental principles and highlight the basic conceptual problems. Some years ago most writers discussing such topics would probably have turned to the example of a ‘particle’ passing through a two-slit apparatus (as in Figure 1.2), whose wave properties are revealed in the interference pattern. Much of the discussion would have been in terms of wave–particle duality and the problems involved in position and momentum measurements, as in the discussion of the uncertainty principle in the last chapter. However, there are essentially an infinite number of places where the particle can be and an infinite number of possible values of its momentum, and this complicates the discussion considerably. We can illustrate all the points of principle we want to discuss by considering situations where a measurement has only a small number of possible outcomes. One such quantity relating to the physics of light beams and photons is known as polarisation . In the next section we discuss it in the context of the classical wave theory of light, and the rest of the chapter extends the concept to situations where the photon nature of light is important.
The previous chapter surveyed part of the rich variety of physical phenomena that can be understood using the ideas of quantum physics. Now that we are beginning the task of looking more deeply into the subject we shall find it useful to concentrate on examples that are comparatively simple to understand but which still illustrate the fundamental principles and highlight the basic conceptual problems. Some years ago most writers discussing such topics would probably have turned to the example of a ‘particle’ passing through a two-slit apparatus (as in Figure 1.2), whose wave properties are revealed in the interference pattern. Much of the discussion would have been in terms of wave–particle duality and the problems involved in position and momentum measurements, as in the discussion of the uncertainty principle in the last chapter. However, there are essentially an infinite number of places where the particle can be and an infinite number of possible values of its momentum, and this complicates the discussion considerably. We can illustrate all the points of principle we want to discuss by considering situations where a measurement has only a small number of possible outcomes. One such quantity relating to the physics of light beams and photons is known as polarisation . In the next section we discuss it in the context of the classical wave theory of light, and the rest of the chapter extends the concept to situations where the photon nature of light is important.
If we turn down our light source from 100 million whatever photons a second to, say, just ten a second, we see the problem, if not the screen, very clearly. These last few photons of light head off, one by one, for the screen, some of them hitting the piece of card but the odd one going through a slit. Now, according to common sense, each should be able to land on the screen unaffected by whether the other slit is there-or not. But none of them, no matter how long we wait, will try to land on the dark areas seen in the interference pattern with lots of light. After all, if an individual photon could behave in this way, it might take a long time but, eventually, the screen would record a picture more like the one-slit experiment with the light diffusing evenly, than the interference one. And the individual photons recognise the contradiction. They continue to behave individually as ‘waves’ all on their own. Points where the light particle could go when just one slit is open become unavailable when the other slit is open. (Incidentally, the same thing, scientists have observed, will happen with any small particle-electrons, whole atoms or even molecules will do just as well.) In experiments, the individual particles of light seem to know about the other slit on the piece of card, indeed what is going on elsewhere in the universe. For, in principle at least, the distance between the slits can be any size, and the same interference effect will apply. (Although if the slits were even a metre apart, the screen might need to be on the moon before the deflected light begins to converge.) Suppose we asked our neighbour if we could borrow their particle detector to check whether the light energy is still travelling in ‘whole photons’, rather than splitting into two and going through both slits and thereby getting in and out of phase with each other. We would find, indeed, that it was still travelling as a whole photon, and that this attempted explanation does not fit with observation. But we would find something else too. The mere act of putting the particle detector next to one slit causes the wavy pattern, the ‘interference’ effect, to disappear!