## Objectives

*After working through this chapter, you should be able to:*

Explain the key features of a wave (wavelength, phase, amplitude)

Explain the difference between phase difference and path difference

Define the terms ‘superposition’, ‘interference’ and ‘diffraction’

Explain what a diffraction pattern looks like

Explain real-world examples of interference to your friends

## Introduction

At this point in the textbook, we have almost exclusively considered light to be travelling in straight lines, and we’ve been discussing principles of **geometric optics** . However, as we touched on in chapter 1 , some phenomena can only be explained by thinking of light as a wave, which is why we consider light to exhibit **wave-particle duality** . This principle highlights that it’s important for us to start thinking about the wave-like properties of light **(physical optics** ). This chapter aims to take you through the fundamental principles of physical optics and introduce **superposition** , **interference** and **diffraction** .

## Features of a single wave

To start with, let’s review what makes up a wave and some of the key terminologies. Fig. 10.1 shows the **peaks/crests** as the top of the wave, with the **troughs** being the bottom of the wave. It also shows the difference between the **wavelength** (distance between two corresponding points on a wave), **amplitude** (maximum distance from the equilibrium point (centre line) to the peak or trough of a wave), and **frequency** (number of **cycles** per second).

As a general rule, the energy of a wave is related to the wavelength (λ) and the amplitude (A). As we learned in chapter 1 , as wavelength increases, energy decreases, and as wavelength decreases, energy increases ( Equation 10.1 ), showing that wavelength and energy have an inverse relationship (as one gets bigger, the other gets smaller).

In terms of amplitude, however, the rules are a little more straightforward, because as amplitude increases, energy also increases. In fact, as the amplitude is doubled, the energy of the wave will be quadrupled, as shown by Equation 10.2 .

The final feature of a wave to discuss is the phase – defined as the location of a point on the wave within a cycle. I find this is made clearer if we imagine the wave is a point rolling on the edge of a wheel, like shown in Fig. 10.2 . Here you can see that at the peak of this wave, its phase corresponds to 90°, and as half a wavelength corresponds to half a phase cycle, the bottom of the trough will be 180° further round, making it correspond to a phase of 270°.

## Features of multiple waves

What happens if more than one wave exists within the same space? We know that white light from the sun, for example, is made up of all the wavelengths of visible light, and we know that we could shine two laser lights at the same spot on a wall (if we felt so inclined) – so what happens to the waves when they meet? Well, when two waves meet each other, they will overlap and interact in a process called **superposition** . This means that the waves will kind of mix together to form a **resultant wave** – and this resultant wave will have a larger or smaller amplitude than the individual waves, depending on how they interact. Fig. 10.3 shows an example where two smaller-amplitude waves (blue) are interacting to form a resultant wave (pink).

Now, let’s go back to think about our extremely common example of shining two lasers at the same spot on a wall. The cool feature of lasers is that they possess only a single wavelength. This means that a red laser, for example, will only output light of a single wavelength in the ‘red’ end of the spectrum (around 700 nm). If we have two identical lasers then, the waves produced will be identical in their wavelength (and therefore frequency), which would make them **coherent** . If we shine one laser at a point on the wall, depending on the distance, it will arrive at a specified point (phase) of the cycle ( Fig. 10.4 A). If we add another laser and shine it at the same point on the wall, and if it is the exact same distance away, it will arrive **in phase** (at the same point in the cycle; Fig. 10.4 B). However, if the distance of the second laser pointer from the wall is varied, then the second wave will arrive at a different point in the phase and will be **out of phase** with the first wave ( Fig. 10.4 C).

For waves to be considered to be **in phase** with one another, they either need to have travelled exactly the same distance or have travelled a distance that results in a 0° or 360° separation. If the waves travel any other distance, resulting in any other degree of phase separation, then they can be described as being out of phase, and the amount of separation between the waves is called the **phase difference** .

Importantly, if multiple waves meet and interact with one another, then they produce **interference** , defined as the variation in wave amplitude that occurs when multiple waves interact. Let’s consider a hypothetical example in which we have two identical light sources producing two identical waves (like with the laser pointers before). If the two waves are in phase, then the amplitudes will add together to increase the amplitude of the resultant wave, which is considered **complete constructive interference** (I like to remember this by thinking that it ‘constructs’ a taller amplitude) ( Fig. 10.5 A) and will produce a bright light ( **maxima** ). However, if the waves are 180° out of phase (meaning the peaks of one wave line up with the troughs of another wave) then the amplitudes will add together to produce a net amplitude of zero, meaning they cancel each other out. This is called **complete destructive interference** ( Fig. 10.5 B) and will produce an absence of light ( **minima** ).

When we talk about waves, we can discuss their phase difference (as before) or their path difference. This means that whilst phase difference can be expressed in degrees of separation in phase, **path difference** is defined as the difference in distance travelled between the two waves and is therefore expressed relative to the wavelength (λ). For constructive interference to occur, we know that we need the waves to be in phase, which means that either they need to travel the same distance (path difference 0λ), or they need to travel a distance that equates to a whole multiple of the wavelength (path difference 1λ, 2λ, 3λ, … nλ). However, with destructive interference, we know the waves will be 180° out of phase, meaning that one will have travelled half a wavelength further (path difference 0.5λ) or any multiple of this that isn’t a whole number (path difference 1.5λ, 2.5λ, 3.5λ, … n+0.5λ). This means that if you slowly increase the path difference between two waves, it will cycle through constructive interference (0λ), destructive interference (0.5λ), constructive interference (1λ), and so on. The exciting part of this is that it means if we have two coherent light sources (identical wavelength and frequency) then we can utilise the amount of interference to measure small differences in distance.

## Utilising interference to measure distances

Let’s imagine we have our two hypothetical, identical light sources again, both producing identical waves of light. If we asked someone to shine them at a wall so that they interfered with one another, and then measured the resultant wave at the wall, we would be able to tell whether they were producing constructive or destructive interference and therefore be able to identify whether there was a path difference. In a classic experimental set-up, the ‘Michelson interferometer’, a single laser light source is shone at a beam splitter to produce two, identical waves. These waves then reflect back off mirrors to meet at a detector which can measure the amount of interference ( Fig. 10.6 ). If you move one of the mirrors by a distance that equates to 0.25 of the wavelength of light, then the wave that reflects off that mirror will have to travel 0.5λ further than the other to reach the detector (0.25λ more to reach the mirror and then 0.25λ more to travel back). This would produce destructive interference at the detector. Now, imagine if the stationary mirror in this set-up was at a distance we didn’t know – for example, if we were wanting to measure the distance between the front of an eye to the retina (axial length). By moving the second mirror back and forth, the interference pattern at the detector will change relative to the distance the light has travelled to the back of the eye, and hence we can determine the distance (in this case axial length). This (in very simple terms) is the fundamental principle of how optical coherence tomography (OCT) works, which will be discussed in chapter 17 .