So far we have thought of light as being made up of little dot-like particles (photons) that zip around in straight lines, changing direction only when they hit something (an object, a refracting surface, or a mirror). The straight paths followed by these little particles are what we think of as light rays. However, this is an incomplete, if not in fact completely wrong, view of what light really is. Photons are not tiny bits of stuff, but something else altogether. What exactly they are is probably beyond our imagining, but we can nonetheless describe their behaviour - incredibly accurately - by a set of mathematical equations known as Quantum Mechanics.
These equations show that while photons sometimes behave like particles, they can also behave like waves, which travel in a definite direction, but are spread-out in space. This wave-like nature of photons leads to a number of interesting and occasionally useful phenomena.
Here we will first introduce the terms needed to describe waves. Then we'll look at some wave phenomena that are directly relevant to optics. Wave optics is complex, and what is given in this chapter is a hugely simplified version of it.
When thinking of light as waves, it is usually enough to imagine it as something like ripples on a pond. Ripples are undulations of water in a gravitational field; the crests of the ripples have high gravitational energy, since they are higher than the surrounding water, while the troughs have lower gravitational energy.
Light waves are also undulations, but they are undulations of an electromagnetic field. We can imagine these undulations looking something like Figure 1. These waves have a number of properties:
These quantities are connected. Imagine two different sets of waves whose peaks travel at the same speed, but one set of waves has a longer wavelength, so the peaks are further apart. Then the frequency of those waves will be lower, because even though their peaks pass us at the same speed, there are fewer peaks for the waves with the longer wavelength. Thus frequency is reduced when wavelength is increased, if the speed stays the same. This can be summarized as a simple equation:
In empty space, all light waves have exactly the same speed (the speed of light) which is about 300,000,000 metres per second, or roughly a million times faster than the speed of sound. The wavelengths of visible light range from 400 to 700 nanometres (a nanometre is one billionth of a metre, so the wavelengths are from 0.0004 to 0.0007 millimetres). Using the above formula, that means the frequencies range from around \( 7 \times 10^{14} \) to \( 4 \times 10^{14} \) Hertz (to compare, sound frequencies range from 20 to 20,000 Hertz).
The speed of a light wave depends on the medium it is travelling through. Light is fastest in empty space, but slows down in water, and is slower still in glass. The refractive index of a substance is just how much slower light travels in the substance compared to empty space:
The number on top of this fraction (\( 3\times 10^8\text{m s}^{-1} \)) is the speed of light in empty space. The change in the speed of a light wave as it moves from one medium to another is what is ultimately responsible for refraction, as we shall see later.
If light is really a wave, what are light rays? The direction that a set of waves are travelling can be thought of as the direction of light rays. Some examples are shown in Figure 2. The rays in this diagram point in the direction of travel, but are perpendicular to the waves.
(b) The top-right image shows a set of straight waves travelling left-to-right. The rays are drawn on top. These waves correspond to parallel light, because the rays are parallel.
(c) The bottom left image shows a fan of circular waves diverging from a point. The rays are drawn on top.
(d) The bottom-right image shows a fan of circular waves converging to a point.
When the waves have a simple pattern (like those in Figure 2 above), it is pretty simple to guess where the waves will be a moment or so from now. At other times, when the waves have a more complex shape, or when they pass by an object, it is not so obvious. In these cases, we can use the Huygens-Fresnel principle to work out where the wave crests go to after a small interval of time \( \Delta t \) has passed. The idea is to imagine each point on a wave crest causing a circular ripple, as shown in Figure 3. Each circular ripple has a radius equal to the distance the wave would travel in the time available, namely \( \text{speed}\times\Delta t \). The new wave crest appears at the front of all the ripples.
Wave crests are an example of what's called a wavefront. A wavefront is a line or curve joining all points in the wave that are either all crests, or all troughs, or some intermediate position between the crest or trough.
Refraction is a consequence of two things: light being a wave, and the speed of light varying depending on the medium it travels through. Figure 4 (a) below is a movie showing what happens when a set of waves travels into a denser medium (in blue), where the speed is less. As each crest goes into the denser medium, it slows down, so the waves "pile up" a little and the wavelength gets less. The frequency is unchanged.
Figure 4 (b) is a movie showing what happens when waves enter a denser medium at an angle. Here, part of the wave (at the bottom of the movie) slows down before the rest of the wave. This makes the wave crest drag behind, and the waves end up changing direction (Recall from [Figure 2(b)]9#Figure2) that the direction of the rays is perpendicular to the wave crests.)
Finally, Figure 4(c) shows what happens when waves hit a curved surface. The parts of the wave hitting the middle of the curve get slowed down before the waves at the edges, and so the entire wave crest gets curved. This then converges. Notice that the waves don't converge to a single point, but remain slightly spread out.
When a series of wave crests enters a denser medium, they slow down. They also get closer together, as well, much like a line of cars entering a road with a lower speed limit bunches up. The waves get closer together in such a way that the frequency never changes. For example, if a light wave with a speed of \( 3\times 10^8\text{ms}^{-1} \), and frequency of 600nm, enters a glass block with a refractive index of 1.5, then the speed drops to \( 3/1.5 = 2\times 10^8\text{ms}^{-1} \), and the wavelength likewise drops to \( 600/1.5 = 400 \)nm.
Since the frequency never changes, it must be somewhat special. In fact, the frequency relates to the energy of the light waves - the higher the frequency, the higher the energy. The fact that frequency doesn't change is simply another way of saying that energy is conserved (it can neither be created or destroyed).
So far we've considered only one set of light waves. But what happens when two sets of light waves meet? It turns out that they can interfere with each other, sometimes cancelling out and sometimes reinforcing. Figure 5 shows what can happen.
When the two waves in Figure 5 don't overlap, there is no interference. However, if you drag the bottom wave to overlap the top wave, different patterns can emerge.
Destructive interference occurs when the troughs of one wave lining up with the peaks of the other. However, it is sometimes more useful to think of one wave as being shifted compared to another. In Figure 5 , to get destructive interference, you have to shift the movable wave half a wavelength compared to the top wave. (A whole wavelength is from peak to peak, so a half wavelength is from peak to trough.) If we shift a wave by half a wavelength, then, its peaks are now where its troughs were, and vice versa. If we shift it by one and a half wavelengths, we also get destructive interference; likewise \( 2 \frac{1}{2} \), \( 3 \frac{1}{2} \) wavelengths, and so on. This is because each shift of a whole number of wavelengths brings the peaks of each wave into alignment, and the extra \( \frac{1}{2} \) wavelength then brings the peak and trough into alignment.
Reflection is a problem for spectacles. Not only do reflections off of spectacles look ugly, they reduce the clarity of the spectacles, since any light that is reflected obviously can't make it through the spectacles into the patient's eye. It would be useful if we could reduce the amount of reflection off the spectacle surface.
We can, by applying an anti-reflection coating to the glass. This coating exploits the phenomenon of destructive interference to reduce the reflection from spectacles. An anti-reflection coating actually creates two reflections. The first reflection is off the anti-reflection coating itself, and the second reflection is from the glass beneath it. The thickness of the coating, however, is arranged so the wave peaks of the first reflection line up with the wave troughs of the second reflection, and so the two reflections interfere destructively and cancel out.
This can be done by making the anti-reflection coating exactly \( \frac{1}{4} \) of a wavelength thick. Then the second reflected wave has to travel an extra \( \frac{1}{2} \) wavelength compared to the first: \( \frac{1}{4} \) through the anti-reflection coating to the glass, then an extra \( \frac{1}{4} \) wavelength back through the anti-reflection coating again before it meets the first reflected wave. Thus the second wave has been shifted \( \frac{1}{2} \) a wavelength compared to the first, so it destructively interferes with it.
Now, \( \frac{1}{4} \) of a wavelength is not very thick, but we can also create an antireflection coating that is \( \frac{3}{4} \) or \( \frac{5}{4} \) or \( \frac{7}{4} \) of a wavelength thick, because these will mean the light wave reflected off the glass is shifted \( 1 \frac{1}{2} \), \( 2 \frac{1}{2} \) , or \( 3 \frac{1}{2} \) wavelengths compared to the wave reflected off the coating, all of which cause destructive interference.
The top wave passes through the anti-reflection coating into the glass. In doing so, two waves are reflected. The first wave reflects off the anti-reflection coating, and the second wave reflects off the glass surface. The reflected waves have been shifted vertically for clarity. Since the coating is \( 1/4 \) of a wavelength thick, the second reflected wave has travelled an extra \( 1/2 \) wavelength compared to the first reflected wave. out.
Just a note: in the above discussion, one thing was left out to simplify the explanation. It turns out that when light is reflected, it also shifts by \( \frac{1}{2} \) of a wavelength. This doesn't actually change anything, since we are trying to create destructive interference between two reflected rays, each of which has been shifted by the same amount when they got reflected.
Diffraction is caused when light waves travel through a narrow hole, or go around a small object. If we thought of light solely in terms of rays, then when light passed through a narrow hole, some of the light would just be blocked, and the rest would travel in the same direction as before However, if we work out what happens to a light wave as it travels through a narrow hole, using the Hygens-Fresnel principle, we find that the light wave spreads out as it goes through the hole.
If you click the arrow at the right, you can see a movie of a wave being diffracted.
If you click the arrow at the right, you can see a movie of waves going around a small object.
The amount of spread caused by diffraction is related to the wavelength of the light waves and to the width of the hole. If we place an image screen behind a small hole, and then shine light through the hole, the light forms a complex diffraction pattern of concentric rings (Figure 6) The pattern is a result of diffraction, which spreads the light out, and interference. The interference is a result of light on one side of the hole interfering with light on the other side. Most of the light, however, is concentrated in the central blob of the diffraction pattern, which is called the Airy Disc.
The angular width of the Airy disc is given, in radians (1 radian \( \approx \) 57 degrees), by the formula
From the formula, the Airy disc is large when the hole diameter is small, but becomes small when the hole diameter is large. The Airy disc is relevant to human vision, because the light focused by the cornea must pass through a small hole - the pupil. We might expect diffraction effects to occur in human vision if the pupil is small enough. Indeed, the angular width of the Airy disc for light with wavelength \( 550 \)nm going through a 1 mm pupil is
This is \( 0.0013\times57=0.076 \) degrees, or about \( 4\frac{1}{2} \) minutes of arc (1 degree = 60 minutes of arc), which is nearly the size of a letter on the 6/6 line of the Snellen chart. This is one reason why our pupil rarely contracts down to 1mm, because the diffraction causes too much blur.
