Chapter 3 Waves

  Chapter 2 explored the idea of light as consisting of fundamental particles, moving along well-defined trajectories. This 『billiard ball』 model, in which a light beam is a collection of individual,compact, and well-localized bundles of energy, stands in contrast to the alternative view, which is that light is a wave. This conception of light has been pursued in parallel with the particle view of light, although it took many years of discussion and experimentation for the wave picture to become fully accepted.

  Unexplained phenomena?

  Other observations, such as those made by Francesco Grimaldi in the mid-17th century, gradually accrued evidence that did not fit with the particle model. Grimaldi saw that light rays deviated from straight lines when they passed through small apertures,such as a tiny hole in a screen. He noted that the light was diffused and that the edges of the beam were fringed with colour, especially pronounced for small objects such as a hair or a piece of gauze. He concluded that the striations seen when light was incident on a small or narrow object were evidence that the light had been bent from its original path as it passed the edges of these objects. If light really consisted of particles moving along straight rays, such solid objects would surely just cast a shadow, and not cause light particles to deviate into strange patterns.

  Further, the problem well known to Newton and his contemporaries of the bizarre way in which light was refracted through certain materials, notably crystals such as calcite—a naturally occurring mineral—confounded explanation in terms of simple particles. An example of this behaviour is shown in Figure 15. The word LIGHT, written on a sheet of paper, is illuminated by a light bulb. Two pieces of calcite have been placed over each half of the writing. In the left half of Figure15(a), two images of the word appear, displaced with respect to each other. In the right half, the two images are displaced in the opposite direction. The lower image in the left half and the upper in the right are just as one would expect from ordinary refraction of light reflected from the paper, seen through the crystal. But the second set of images appears to arise as if from a different refractive index. By placing a polarizer over the crystal, as shown in Figure 15(b)and 15(c), it is possible to isolate images formed by two different orientations of polarized light. Each polarization experiences a different refractive index. This is the phenomenon of birefringence.

  15. An illustration of birefringence. The image of the word LIGHT viewed through a pair of calcite crystals, using a. unpolarized light, b. vertically polarized light, and c. horizontally polarized light.

  These observations all pointed to features of light that demanded explanation. They are known respectively as interference,diffraction, and polarization. We will explore these phenomena in this chapter, developing a picture of light as wave motion.

  Wavelength and wave frequency

  What are the characteristics of wave motion? Waves are a form of undulation associated with a medium, for example water waves on the surface of a pond. These waves consist of the up-and-down motion of water molecules at the interface between the liquid and the air. The highest and lowest points of this motion become the peaks and troughs of the water wave, while the wave itself moves across the surface—that is, at right angles to the motion of the water molecules. For this reason it is called a transverse wave. Its speed depends on the depth of the water, among other things.

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  The circular surface waves that radiate away from the point at which a stone is cast into the water, as shown in Figure 16a, are a familiar effect. The distance between successive peaks is called the wavelength (Figure 16b), and the rate at which peaks hit the shore is called the frequency (Figure 16c). The product of these two quantities is the speed of the wave.

  The puzzle for many centuries was what kind of undulations constituted light. It was assumed that some medium would need to exist in order to support a wave. And since the speed of light is so large, this would need to be a very stiff medium. But a stiff medium would surely make it difficult for other bodies to move through it. For instance, because we can see distant stars, there should surely be some medium to support the propagation of light between the Earth and the star. Since the Earth moves around the Sun, the planet would be continually swept by a 『wind』as it travelled through such a medium. This enigmatic medium was called the 『aether』, and it was not until the end of the19th century that it was finally discarded as a useful concept.

  16. A circular wave on a water surface: a. contours of equal height, called wavefronts, b. wave height vs distance from centre at a particular instant, c. wave height vs time at a particular location.

  So, what sort of wave is light? This was also finally answered in the19th century by James Clerk Maxwell, who showed that it is an oscillation of a new kind of entity: the electromagnetic field. This field is effectively a force that acts on electric charges and magnetic materials. For example, a cloth charged with static electricity will attract dust particles to it. A magnet will be attracted to the door of a refrigerator. In the latter case you can feel this force as you place the magnet close to the door: the magnet accelerates towards the door unless an opposing force is supplied.

  In both cases, a force pulls one object to the other, so that at some distance away from the cloth the dust particle feels this force due to the electric field generated by the charges on the cloth. And similarly for the refrigerator door, due to the magnetic field generated by the magnet. In the early 19th century, Michael Faraday had shown the close connections between electric and magnetic fields. Maxwell brought them together, as the electromagnetic force field. It turns out that in the wave model,light can be considered as very high frequency oscillations of the electromagnetic field. One consequence of this idea is that moving electric charges can generate light waves. I』ll discuss this, and other methods for making light, in Chapter 5.

  Interference

  If two stones are dropped in the water next to one another, the resulting two sets of expanding circular waves collide. Some of the peaks are therefore higher, where the waves reinforce each other. But there are also lines of flat water, where there is no up-and-down movement of the water surface, even though the waves generated by both stones』 disturbance of the water are passing through all points along these lines. The location of such lines is shown in Figure 17, along with the wavefronts—the locus of points on the peaks of the waves emanating from each source.

  17. a. Two interfering circular waves on a water surface. The dashed lines are contours of equal phase. b. Constructive interference between paths of equal length. c. Destructive interference between paths difering by half a wavelength.

  This phenomenon is known as interference, and it arises from the addition of the amplitude of two waves when they meet. If the peaks of the two waves coincide, the resulting peak is twice the size, or amplitude, of either wave. The waves are said to be in phase, and the interference is constructive. This is shown in the middle diagram of Figure 17. If, however, the waves are exactly out of phase, so that the peaks of one coincide with the troughs of the other, then the resultant wave has an amplitude of zero: the waves have 『cancelled』 one another or interfered destructively. This is shown in the bottom diagram of Figure 17. It is immediately clear that such a phenomenon could not happen with particles, for how could two particles cancel one another?

  It was the observation of interference effects in a famous experiment by Thomas Young in 1803 that really put the wave picture of light as the leading candidate as an explanation of the nature of light. Young’s experiment was very simple, and very elegant. He took a candle as a light source, and placed it behind a screen in which there were two holes a very small distance apart. The light shining through these holes could be seen on a second screen some distance away. Now, if only a single hole was used—say by covering up the second—then the light made a single spot on the screen. When both holes were open, however,something marvellous happened—instead of just a spot of twice the brightness, the spot now had striations. These are generally nearly straight lines of zero intensity perpendicular to the line joining the centre of the two spots, as shown in cross-section in Figure 18 arising from the interference of the waves passing through the two holes. They are called 『Young’s fringes』, and are one of the key pieces of evidence for light as a wave motion.

  How does interference explain the coloured 『fringes』 observed by Newton in reflections from two surfaces very close together? What is needed for interference to occur is two waves, the relative phase of which (that is, the relative positions of the peaks of the two waves) can be adjusted. In Newton’s experiment, interference occurs because a single incident beam of light is partially reflected at the two surfaces and thus split into two waves. If the distance between the two surfaces is equal to a single wavelength of the light, then the peaks of the two reflected waves coincide, and there is a bright fringe. If, however, the distance is just half of the wavelength, then the peak of one lines up with the trough of the second and there is a 『dark fringe』 owing to destructive interference. Thus, when you look at the surface the bright and dark fringes map out the contours of the separation of the surface with a resolution of less than one wavelength. For green light, with a wavelength of approximately one-half of a millionth of a metre,or 500 nm, the separation can be determined to better than 250nm—a precision of about 1/40th of the diameter of a human hair.

  18. Thomas Young’s experiment. The light from one slit produces asmooth intensity pattern. When both slits are open 『fringes』 appear,characteristic of wave-like behaviour.

  Of course, for a different wavelength the bright and dark fringes will occur in a different place, so that the surface shows fringes of colour when illuminated with white light. It is interference of light waves that causes the colours in a thin film of oil floating on water.

  Interference transforms very small distances, on the order of the wavelength of light, into very big changes in light intensity—from no light to four times as bright as the individual constituent waves. Such changes in intensity are easy to detect or see, and thus interference is a very good way to measure small changes in displacement on the scale of the wavelength of light. Many optical sensors are based on interference effects.

  Holography

  Interference is also the means by which one can make true 3D images, that is, images that can be viewed from different angles and reveal different aspects of the object. These are different to the synthetic images in so-called 3D films, and are called holograms. Holograms are made by recording the full waveform of light scattered from an object. The sort of 2D images we’re used to from photography encode only the amplitude of the waves. The phase information is lost. This is because detectors only respond to amplitude, so there is no way in the sorts of images we have looked at previously to extract phase. Nonetheless, it is the phase of the wavefronts scattered from an object that encodes its shape.

  What interference enables is the encoding of phase into intensity,so that photodetectors can register patterns in which the full amplitude and phase information of the object wave is recorded.The principle is shown in Figure 19. The wave scattered from the object interferes with a reference wave that has a well-known shape, often generated by a laser. The interference pattern is recorded on a detector or in light-sensitive materials: this is the hologram itself, invented by Denis Gabor in the middle part of the 20th century.

  19. A hologram is constructed by recording the interference fringes between a reference beam of light and one scattered from the object.

  Viewing the image is a little more complicated than with an ordinary photograph. First, a reference beam illuminates the hologram and some of its light is scattered from the patterns encoded in the material. These scattered beams have the remarkable property that they reproduce the beams scattered from the original object, so that as your eyes receive them, it appears that the object is reconstructed in front of you. Moving around the image reveals different aspects of the object, because the beams scattered from those different parts encode different information.

  Holograms can also be computer generated and embossed in metals or other materials. The idea is that the surface shape mimics the interference pattern of the reference and object waves,so that raised parts correspond to bright fringes and low parts to dark fringes. Again, illuminating this with a reference wave causes scattered light to imitate the wavefronts of the chosen object.This kind of hologram is used as a security device, including on some banknotes (such as the UK ￡20 note, which has a strip of holographic images of the 18th century Scottish economist Adam Smith on it), because they are hard to make and require advanced engineering to copy.

  Limits to imaging, redux

  The wave picture of light also explains why we can’t see infinitesimally small objects using a microscope, just as Abbe realized. Very tiny things—down to about half of one micron (one millionth of a metre, or about half a wavelength of visible light)can be seen using an ordinary light microscope. Much more sophisticated methods are required to see even smaller things.The reason is that the wave nature of light puts a lower limit on the size of a spot of light.

  I noted previously that when two light beams meet, they interfere to give regions of zero amplitude—dark fringes. The distance between these fringes is actually determined by the angle at which the two beams intersect. If the angle is very large, the fringe spacing is small; if the angle is small, the fringe spacing is larger.The smallest possible separation of the fringes is one wavelength,or about one micron for visible light.

  Now, if this fringe pattern is recorded as a hologram, then when it is illuminated again with the reference beam two beams will emerge in the directions of the light beams that were used to record the interference pattern. In order to see such a fringe pattern with a microscope the lens must capture both of these beams to form an image of the fringes. If this didn’t happen, say because the lens could only capture one beam, then there would be no interference fringes in the image.

  This is the physical basis of Abbe’s criterion that I introduced in Chapter 2: the maximum angle between two beams captured by the lens of an imaging system sets the minimum object size that can be resolved. It’s straightforward to see then that the best any lens system can do is to see object sizes of about the wavelength of the illumination. Thus conventional light microscopes can see very tiny objects, about fifty times smaller than a human hair,but not much smaller than this. They can be used to look at biological cells, but not the cell nuclei, for instance.

  Super-resolution imaging

  Optical scientists and engineers have found a number of very clever ways to get around the object size limits posed by conventional microscopes, so that they can see inside cells, or view objects that are more than one hundred times smaller than the wavelength of light. These instruments make use of new materials and processes, such as the ability to attach nanometre-scale particles to the objects you wish to see, or to insert molecules that emit light into cells. These emit light (they fluoresce) at a long wavelength when they are illuminated by a beam of light at shorter wavelengths. Since they are much smaller than the resolution of the microscope lens, the resulting image is a spot the size of which is entirely limited by the microscope optics, according to Abbe’s formula. But the exact centre of this image can be located very precisely by using a camera to take a long look at the fluorescence of the attached nanoparticles and determine the position of the maximum intensity of the spot. This approach is called photo-activated localization microscopy, or PALM, and was invented by Eric Betzig in the US. It has revolutionized live cell imaging, allowing faster acquisition and precise depth resolution over a wide field of view.

  Another method for measuring small structures in larger fluorescent objects is to shine a second annular beam on them that causes the objects illuminated by this beam to have their fluorescence extinguished, so that the remaining fluorescence can be located precisely by the same approach as described previously.This method is called stimulated emission depletion microscopy,or STED, and was invented by Stefan Hell in Germany. I will describe the process of stimulated emission in more detail in Chapter 5. These novel ways of making high-resolution images that enable intracellular structures to be imaged have had enormous impact in biology and medicine. Indeed the significance of this impact was recognized when the 2014 Nobel Prize in Chemistry was awarded to Betzig and Hell.

  The process identified by Abbe works in reverse, too. When applied to the illumination of a sample by means of a microscope lens, it says that a light beam cannot be focused to a spot much smaller in diameter than a wavelength. Again, the tightness of the focus depends on the range of angles that the lens can produce on the side that faces towards the object: the broader the range of ray directions, the tighter the focus of the light.

  The relationship between the range of angles between interfering beams and the size of the fringe structures turns out to be a very fundamental property of waves. This idea was quantified by Joseph Fourier, a French scientist of the early 19th century, who provided a detailed mathematical analysis of light wave propagation.Fourier’s theorem says in simple terms that the smaller you focus light, the broader the range of wave directions you need to achieve this spot.

  Diffraction

  This explains another feature of light beams—they gradually diverge as they propagate. This is because a beam of light, which by definition has a limited spatial extent, must be made up of waves that propagate in more than one direction. The idea can be tested using a laser pointer. The beam emitted by the laser itself is about 10 μm (ten millionths of a metre) or so in diameter. When it reaches the screen, it is about 1 mm (a thousandth of a metre)in diameter. And if it were sent further away, say to the Moon(approximately 400,000 km), then it would be about 24 km in diameter! This phenomenon is called diffraction.

  Diffraction has some interesting applications in determining the shape and symmetry of structures. For instance, when you shine a beam of light on a screen with small holes in it, of diameter comparable to the wavelength of the light, the light diffracts through the apertures, spreading in inverse proportion to the size of the aperture. These diffracted beams interfere some distance away from the screen, and the resulting interference fringes—the so-called diffraction pattern—tell something about the size and relative location of the apertures. For instance, if they are in a regular array, then the diffraction pattern will also show regularities. The advantage of using such patterns to measure this type of object is that you don’t need to have very expensive or complicated lens systems or detectors close to the object—you simply look at the pattern when it has naturally expanded due to diffraction.

  Now, imagine that the screen is replaced by a transparent solid material, say a crystallized protein structure. The 『holes』 are replaced by the atoms in the protein molecule, which are very small indeed and are connected to one another by bonds in the molecule that are about one tenth of a billionth of a metre (0.1 nm) in length.If light with a wavelength of about this size illuminates such a structure, then the light will be diffracted. The actual structure of the molecule itself can be determined from the diffraction pattern.This is the basis of X-ray diffraction. As noted in Chapter 1, it was famously used in the effort to find the structure of DNA and is now a very common tool in biochemistry, used regularly for finding out the structure of new molecules that might be useful in developing drugs for example. It requires a bright X-ray light source, as well as a means to make crystals out of the molecules. Figure 20 shows a diffraction pattern from a crystal of bovine enterovirus.

  20. An X-ray difraction pattern of a protein crystal taken using a modern synchrotron X-ray source.

  It’s clear that if you want to transmit light over long distances,then diffraction could be a problem. It will cause the energy in the light beam to spread out, so that you would need a bigger and bigger optical system and detector to capture all of it. This is important for telecommunications, since nearly all of the information transmitted over long-distance communications links is encoded on to light beams.

  Guided waves

  The means to manage diffraction so that long-distance communication is possible is to use wave guides, such as optical fibres. A wave guide is a structure that has a carefully designed profile for the refractive index. For example, the index variation across an optical fibre is arranged so that the 『core』, with a diameter of a few millionths of a metre, has a higher index than the surrounding 『cladding』. The light is confined to this higher index core and moves along the fibre without diffracting,remaining the same size over very long distances—across the Atlantic Ocean via undersea cables, for example. This means of controlling light is the basis of a wide-ranging optical information infrastructure, from communications to sensors.

  Polarization

  The final important feature of the wave model is the characteristic of polarization. Recall that in transverse waves the undulations occur in a direction at right angles to the direction of propagation of the wave. Importantly there are two such directions for these undulations.

  Consider waves on a string, for instance. If you move one end of a rope up and down rapidly, you can see the undulations move along the rope. A similar thing happens if you move the end left and right equally rapidly. Both vertical and horizontal oscillations are at right angles to the motion of the wave down the rope. The waves are said to be 『transverse』.

  Light waves behave similarly. For example, horizontally polarized light has an electric field that oscillates in a horizontal plane(relative to, say, an optical bench). Similarly, the field of a vertically polarized light beam oscillates in a vertical plane. (There are more complicated forms of polarization too, but these will suffice for now.) The phenomenon of birefringence can be explained by noting that a crystal is a structure of atoms in very specific and highly repeatable units. These 『unit cells』, consisting of a few atoms, may themselves be asymmetric. Depending on whether the light is polarized along the long axis or the short axis of the unit cell, it will experience a different refractive index, and thus will be deviated by different amounts as it propagates through a block of such material.

  One of the well-known ways in which polarization of light is exploited is in sunglasses. Some of these use lenses made of plastic material (for example, a sheet of Polaroid plastic) that acts as a polarizer—that is, an element which transmits light of a particular polarization (say vertical) and absorbs that of the orthogonal polarization (horizontal). Polaroid is made from rugby-ball shaped molecules that are aligned and 『frozen』 in the plastic polymer. These molecules preferentially absorb light that is polarized along the axis of alignment of the molecules.Since generally sunlight scattered from objects has a random polarization (roughly 50 per cent of each polarization direction),then filtering out light of one polarization effectively reduces the brightness of the scene by half. Further, polarized sunglasses reduce glare: that is, light reflected from shiny flat surfaces, such as the hood and windscreen of cars. These surfaces tend to reflect a higher proportion of light polarized in a direction parallel to the surface (an effect discovered in the 19th century by Sir David Brewster and now named for him). Such reflections are blocked by spectacles that are constructed in the manner noted here,making for clearer vision of the road.

  Transparent birefringent materials can also change the polarization of light without absorbing the light. This is because the speed of light depends on the direction of polarization with respect to the 『orientation』 of the material. Some materials, such as ordinary glass, have no special orientation: you can rotate the material with no change in the effect on a light beam. As noted above, atoms in birefringent materials are arranged in such a way that there is a preferential direction—the symmetry axis—along which atoms respond differently to light. That is,light polarized along the symmetry axis will go slower (say) than light polarized perpendicular to the axis. Now, imagine that the light is polarized at 45 degrees to the symmetry axis. We can consider that half of this light is then polarized along the symmetry direction and half polarized at right angles to this direction. If the latter is slowed down by a sufficient amount,then the light emerging at the other face of the material can be polarized at -45 degrees. Thus the direction of polarization has been 『rotated』 by 90 degrees.

  Some birefringent materials can be used to control the state of polarization, by actively adjusting the orientation of the alignment axis of the molecules, say, using an applied voltage across the material itself. An example of this is a class of materials called liquid crystals (LCs), which consist of elongated molecules. The orientation of the molecules in an LC can be controlled by applying a voltage across them. Other materials become birefringent when forces or stresses are applied, because the force causes the molecules to rotate or the atoms to change alignment. This phenomenon enables the construction of force sensors by monitoring the state of polarization of light at the output of the optical sensor.

  Placing a piece of birefringent LC between two polarizers also allows control of the light intensity by means of an electrical connection. Applying a voltage reorients the molecules, thus changing the refractive index seen by a polarized light beam. If a polarizer is placed after the LC then depending on the applied voltage a greater or lesser amount of light will be transmitted through the final polarizer. An array of such 『cells』, each driven by a separate electrical signal, can be used to form a display where each cell is a single pixel. This is the basis of a liquid crystal display (LCD), and is often used for computer display screens and television sets.

  In fact, such displays can be used to show 3D movies. The illusion of depth in these movies derives from the stereography of human vision. Each of our eyes sees a scene from a slightly different location, since they are a few centimetres apart in our skulls. The two images are combined in our brains to give us a perception of depth.

  This illusion is reproduced using 3D glasses by means of polarization. Two images are projected on to the display or screen.Each of these is generated using light of a specific polarization,and each is shot from a slightly different vantage point. The 3D glasses are polarizers set at different orientations that allow transmission of one scene to the left eye and one to the right, each completely blocking the alternate image that has the 『wrong』polarization. Thus we have a sense that the scene is as we would perceive it in the natural world—that is, the illusion of a three-dimensional arrangement of objects and people.

  The successes of the wave model of light have been stunning,allowing us to understand some important characteristics of light and to use this understanding to build new technologies. And the successes of the ray picture of light have been equally amazing.Yet it is certainly puzzling that two apparently very different views of what light is should be necessary. It is to this conundrum that I now turn.