# Superposition, interference and diffraction

## 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).

E=hcλ

energy=Plancks constant×speed of lightwavelength of light

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 .

E∝A2

energy(is proportional to)amplitude2

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 ).