The question ‘is light a ray, a wave or a particle?’ has long fascinated philosophers and has intensified since the concept of the photon first emerged at the beginning of the twentieth century. The pragmatic answer to the question is ‘it depends what you are looking for’ and is really about how best to conceptualise light in particular circumstances. We have already mentioned such distinctions briefly in Chapter 1. This chapter will extend these basic concepts to introduce the material which follows in later chapters.
2.1 Light in a Vacuum
This is the simplest case. When no other materials impede its progress or modify its path (Figure 2.1), propagation in a vacuum can be accurately described through a wave propagation model. Ray optics also works since in an idealised infinite vacuum there are no edges or multiple paths through which the light can interfere or diffract.
By the same token, however, a particle of light will also proceed unimpaired along the same path as the ray. And, already, we have an apparent contradiction with the wave propagation model, in that a single photon cannot spread its presence over a surface area increasing as the square of the distance from the source. This dilemma continues to produce philosophical debate but really only becomes pertinent when we produce single identifiable photons and watch them in motion. More on this later!
Perhaps the principal useful aspect of propagation in a vacuum is that light then proceeds at the velocity commonly approximated as c = 3 × 108 m/s.
2.2 Light in Isotropic and Anisotropic Materials
In a material, light, as it propagates, interacts with the molecules within the material, typically by inducing some form of electronic motion within the molecular structure; this may result in dipole moments which change in response to the lightwave’s electric field (Figure 2.2).
The slowing down of light in a material can be viewed as due to the energy storage and release through the electron motion induced in the material, which introduces a delay in propagation. Through similar reasoning (and confirmed through the Clausius–Mossotti equation) the more energy that is stored the larger the delay, so the more molecules per unit volume the higher the refractive index. By the same token, in the case of anisotropic materials, exemplified in many crystalline structures, the refractive index seen by an incoming electromagnetic wave will be higher when its electric field oscillates in a crystalline plane with a higher density of molecules. An electric field oscillating in a plane with lower density will experience a lower refractive index and so will travel faster. (This phenomenon is known as birefringence.) Hence the different polarisation states (Figure 2.4) of an incident light beam with respect to the material’s orientation will see a range of refractive indices.
Another important observation here is that the refractive index for polarisation states other than the two states aligned with the crystalline structure of the material will be a mix of two values – the input state will be split into these two components along the so-called principal axes of the material. Each component will consequently exit with a different phase delay, so that the output polarisation state will in general be different from the input state. Appendix 1 explores this in more detail.
There are many other insights which are enabled by the mass–spring–damper model. Any mass–spring–damper arrangement involves a frequency response to a given applied force. At low frequencies the deflection introduced by the applied force has a particular characteristic value and is in phase with the applied force. At the resonant frequency, the deflection is a maximum but is now in quadrature with the applied force. After passing through the resonance the displacement becomes small and in antiphase (Figure 2.5).
By the same token, then the refractive index will have exactly the same general resonance characteristics, as the mass–spring–damper model though in molecular materials there are many resonances and therefore the refractive index behaviour follows the trends indicated in Figure 2.6. Consequently, since all these resonance effects are taking place, the refractive index does vary with frequency and so the velocity of light within the material is also a function of frequency – a phenomenon known as dispersion. Looking at an everyday example, at low frequencies water has a dielectric constant of 80, corresponding to a refractive index of about 9, but the refractive index in the visible region is about 1.3.
Figure 2.6 also imples a number of other important photonic features, common to all materials. The refractive index generally decreases with frequency except around the resonances, after which there is another step downwards. Also, the refractive index can drop below unity. But does the velocity of light ever exceed its velocity in a vacuum? The implications of this apparent dilemma are examined briefly in Appendix 4, which considers the phase and group velocity concepts.
Thus far we have ignored another important aspect of our mass–spring–damper model – namely, the damper. This represents the sources of loss, and this lost energy is converted typically into heat. So, strictly speaking, the refractive index, which describes the transfer of energy through the material, should include this loss term. This is expressed through the representation n′ = n + jk, where j is the square root of −1 and k is the known as the imaginary component of the refractive index. Additionally, as indicated in Figure 2.6, the loss terms increase significantly at resonance since this is where the radiation is more readily absorbed, typically by a large factor. This absorbance feature is one source of the colour in the world around us.The propagation of light within a homogenous material is then in general described through the propagating wave equation
Here E is the electric field amplitude of the propagating wave, z the axis along which the wave propagates, ω the optical angular frequency in radians/second and λ the wavelength in vacuum; and the final term represents the attenuation of the electromagnetic wave. The first term gives the time dependence, and the second term the distance dependence, of the optical phase. Note that the imaginary part k of the refractive index determines the level of attenuation.
There is yet another aspect to this – suppose the electrons are no longer bound to individual molecules but free to move. This happens, for example, in conducting metals and also in the ionosphere around the earth. In this case the restoring force is due to the electrostatic forces exerted by the positive ion lattice, as indicated schematically in Figure 2.7. Remember, too, in the case of conductors, the losses are due to the collisions between the moving electrons and the stationary ions, a phenomenon encapsulated in the electron relaxation time for the material concerned.
This plasma resonance has many interesting ramifications. Everyone is aware that X-rays travel through metal but light does not. Satellite broadcasting in the GHz region comes down from outer space through the ionosphere. However, much lower-frequency radio waves in the short wave band bounce off the ionosphere in the same way that light bounces off metals.
The impact of the relaxation time in plasmas (see below) is also important. As mentioned above, the conduction losses in a metal are basically determined by the frequency of collisions between the freely flowing conduction electrons and the stationary ions in the metal. However, if we apply an electric field at a frequency which exceeds this collision frequency, the inverse of which is the relaxation time, then there are many fewer collisions between the conduction electrons and the stationary particles in the metal, so the losses decrease dramatically – in other words, the metal becomes transparent.
The refractive index of gold as a function of wavelength is indicated in Figure 2.8. Among other things the rapid variation of the complex index with wavelength in the visible region (roughly 0.4–0.8 µm) accounts for the difference in the colour of gold as compared, for example, to the colour of silver. Most metals in and around the optical and ultraviolet region exhibit an intriguing transition between relatively low-frequency behaviour as a conductor and the higher-frequency behaviour as a relatively transparent dielectric. In other words, the electrons in the metal constantly turn around so that there is no current flow and likewise no collisions to cause absorption losses. This fascinating transition has, within the past couple of decades, evolved into so-called plasmonics, which seeks to understand and exploit this somewhat complex evolution.
This discussion has been about light as a wave. There has been no mention of the photon. These classical wave approaches do give a very reliable guide and an indispensable insight into the interaction between light and materials. The photon model also has its place, however. This is most apparent with regard to the energy levels evident in a material (Figure 2.9), a concept which has evolved as the ‘conventional’ approach to explaining the wavelength-selective absorption of light in materials.
The differences between these energy levels and the ground state in a molecule equal the photon energy required to excite the molecule from the stable ground state. This absorbed energy then typically (but not always) re-emerges as heat when the molecule relaxes to its ground state. The same behaviour can be ascribed to the resonant absorption of mass–spring–damper systems if we are looking purely to match the frequency of the input light with the frequency of the mechanical resonance. This merging of the wave (and mechanical resonance) domain and the photon domain (via energy level differences) can provide useful insights into material behaviour.
There are lower-frequency transitions in molecules as well, corresponding to, for example, exciting an entire molecule into rotational rather than electron-orbit resonances (see Figure 2.6 for the general trends). A microwave cooker which operates at around 2.5 GHz exploits exactly this effect, tuning to resonances in water, fats and sugars. Here, however, we are by no means concerned with a photon energy in the order of a few microvolts; this is well below the photonics region discussed in Chapter 1. The scientific community also talks of these frequencies in terms of molecular resonances, implying an analogous electromagnetic concept.
Much of the wave and photon dilemma is encapsulated in the above discussion. Whilst numerous weighty texts have been published on the subject there is really no definitive answer. For us it amounts to finding the approach which gives the most useful model to analyse and understand the photonics scenario of interest.
2.3 Light Interacting with Structures
A material structure, in our present context, is an arrangement which comprises two or more materials with different optical properties. Once again, the question of a structure’s scale compared with the optical wavelength enters the discussion.So, we start with reflection and refraction at a large-scale structure, namely an interface between two dielectric materials (Figure 2.10). We will simplify the discussion by assuming that the interface is planar. The light that passes through the interface is refracted by an amount determined by Snell’s law, namely:
The reflected light obeys the normal ‘angle of incidence equals angle of reflection’ relationship, but what is interesting here is the amount which is reflected and the way in which this amount depends upon the incident angle. Indeed, at one specific angle – the Brewster angle – only one polarisation state, namely that with the electric vector aligned with the surface of the interface (sometimes called S (in-plane) polarisation and sometimes called TE for transverse electric polarisation) is reflected. The reflection coefficient is not 100%, so that the refracted ray still contains some of this polarisation state. The reflection coefficient as a function of angle for the two linear polarisation states is also shown in Figure 2.10. The partial removal of the S polarisation state (with electric field vibrations perpendicular to the P state vibrations and to the direction of the reflected ray) manifests itself in daily life, amongst other things in the polarising sunglasses designed to reject the (partially) polarised reflected light from our surroundings.
The discussion thus far concerns a light beam travelling from a low-index medium to a higher-index medium. When the situation is reversed, there will be a case when the angle of incidence is high enough for refraction to occur into the interface plane (i.e. the angle of refraction is 90°), and then the incident angle is referred to as the critical angle. If the incident angle goes beyond that, total internal reflection occurs, as implied in Figure 2.10, which also indicates some reflection for incident angles below the critical angle.
Incidentally, the derivation of Snell’s law (see the Chapter 2 problems) exemplifies a very important principle in understanding the coupling of light from one material or structure into another. This principle is that the projected components of the wavelengths along the interface must be the same – a concept known as phase matching. This idea applies to any wave propagating through an interface, whether the wave is acoustic, electromagnetic or even a water wave.
There are many other possible interfaces. In the above we have assumed a perfectly planar interface. However, slight roughness will introduce corresponding slight changes in the angles of incidence and refraction until, in the limit, a perfectly scattering surface would send the light in all directions. Again we encounter this daily – sunlight entering your room is scattered throughout the entire volume of the room, by the walls and furniture.Moving on to a medium-scale structure reaches into diffraction and interference: we move from the ray optic approach, which works at large scales, into the wave optics approach. Much is encapsulated in the classic Young’s slits experiment, shown in Figure 2.11. Light from a single-parallel-beam single-wavelength source impinges on two slits, separated by a distance d and each of a width conveniently assumed to be much less than the wavelength. The input light then reradiates in all directions and produces an interference pattern as indicated. The interference pattern is determined by the vector sum of the fields directed through the two paths. Full constructive interference occurs when the path difference dA – dB is an integer number n of wavelengths, which gives, for y » d,
where x is the distance on the screen from the axis of the system and y is the distance along the axis between the slits and the screen. There is a time delay between arrivals at a particular point on the screen from the two paths and there is an implicit assumption in equation 2.5 that the phase delay on the optical signal is completely predictable for this time difference. Sometimes it is not, which leads us into the concept of coherence; ultimately this coherence concept applies to both the spatial and temporal behaviour within an optical source, a subject to which we shall return later.
Much of photonics concerns this medium-scale structure, ranging from camera lenses and image sensing arrays to DVD players and the display on your telephone handset. Furthermore, almost all of this discussion is concerned with wave optics.
At small scales we move into the domain of subwavelength structure in a material. This domain has only recently become available, with the notable exception of materials such as colloidal gold, which has been used in pottery for centuries and for which the colour can range through reds to blue depending upon the diameter of the tiny gold spheres within the colloid. The resonance wavelength for the electron plasma circulating the spheres varies as indicated in Figure 2.12, but gold spheres throughout this dimensional range also absorb more in the blue than the red, owing to material absorption, the property which makes ‘normal’ gold appear golden. In current terminology these gold spheres would be regarded as an example of quantum dots. As colloidal gold they were simply a way of introducing permanent colour into artefacts. The tiny structural dimensions take the ‘normal’ colour far from its golden origins.
Much of the story here is concerned with the ability of structures to produce what are, in effect, wavelength filters. Examples of such phenomena occur extensively in nature – butterfly wings are amongst the most common and striking. This comes initially as something of a surprise: let’s make something a different shape and it will appear as something different. However, a few thoughts on electronic circuits shows how we take this for granted. As far as a radio frequency wave is concerned, a piece of straight wire appears as an object very different from a coil of wire yet both have dimensions significantly less than a wavelength. What has become known as ‘nanophotonics’ has been described as a manifestation of this familiar concept – so here light becomes an electromagnetic wave but in the sense of an electric current.
Is light a wave, a particle or a ray or an electric current? The answer is yes to all four possibilities, but it all depends on the context. The preceding discussion has given a glimpse of all these aspects and exemplified some of the circumstances in which light is more readily perceived using one or the other model. There is some interchangeability but usually an appropriate combination of points of view is the simplest way to arrive at a usable understanding. It is quite a fight, say, to model the refractive index through quantum mechanics and particles, but in general bringing one model into the domain of another, as Nils Bohr did with his hydrogen atom model, can give useful, workable and, with care, accurate results. Of course, those of a philosophical nature should consult examples of the many extensive texts on the nature of light.
These problems could involve collaborative searching on the internet.
1. What features determine the colour of the objects around you? Consider, for example, butterflies, plants, the bloom on a camera lens or some spectacle lenses, the iris of your eye.
2. What are the light sources around you – for lighting in your room, for your DVD player, in your fluorescent watch, in a ‘black lighting’ tee shirt, in the sun, in the TV screen and in those old cathode ray tube TV sets? How do these sources vary in structure, in their physiological effect on the observer, in beam quality etc.?
3. What determines the colours you perceive when looking at a particular object? Why do colours apparently differ when viewed at different times of day, or when one is using different light sources or when one goes abruptly from the dark into a brightly lit room? How, for example, could the texture and colour of a wall make a huge difference to the brightness of a room as sunlight comes into it – what might be the mechanisms at work? You should take into consideration the way in which the eye responds to different colours and levels of light.
4. Think about a planar interface between two materials with refractive indices, n1 and n2, with n1 < n2.
(a) Demonstrate that Snell’s law relating the angles of incidence and refraction from material 1 to material 2 can be understood by considering the projection of the components of the optical wavelengths of the incident and refracted rays along the optical interface.
(b) What is described in part (a) is an example of ‘phase matching’, also sometimes called ‘resonant coupling’. Look around and discuss where this general effect may be taking place, both in everyday events and in the technological world in which we live.
(c) We mentioned the Brewster angle in the text. At this angle, when light traverses a planar interface from a low-index region into a high-index region, the reflection coefficient for the electric field vibrations parallel to the plane of incidence is zero, giving rise to partially polarised light in the reflected beam. Investigate the generalisation of this observation – look up Fresnel reflection on the web – and note not only the changes in reflection amplitude but also the changes in phase.
5. Would a suspension of gold nanospheres (Figure 2.12) appear the same colour in water, air and high-density glassware? If so, why, and if not, why not?
(a) Look up typical values for the free electron density in the ionosphere, in gold and in n-type silicon and use equation 2.3 to arrive at the plasma resonance frequencies.
(b) From this, comment on when and how these materials become transparent to electromagnetic waves and evaluate the significance of your calculations in terms of possible applications.
7. We have indicated in this chapter that there are practical circumstances when the refractive index can be less than 1. This implies a velocity of light which exceeds the velocity of light in vacuum. However, it is widely accepted that the speed of light in a vacuum is the fastest that light can travel. How can we rationalise these two observations? The key lies in the subtleties of phase and group velocities, hinted at in problem 4 above and discussed a little more in Appendix 4.