Objectives
After working through this chapter, you should be able to:
Define the laws of reflection
Calculate the angle of reflection
Understand image formation at a plane mirror
Understand how the equations for spherical mirrors differ from those of refractive surfaces and lenses
Determine linear magnification of an image formed by reflection
Introduction
By this point we’ve covered the basics of refraction, but this is only half the story. Indeed, light can also be reflected by surfaces, and this is an important principle of optical systems. This chapter will focus on reflection at plane (flat) and curved surfaces.
What is reflection?
When we say something is ‘reflected’, we mean that it is returned in the plane it originally came from. The principle itself can be likened to bouncing a ball at a wall (providing we pretend gravity isn’t a factor) – the ball cannot travel through the wall, so instead it will bounce back as a specified angle depending on the angle it approached the wall ( Fig. 6.1 ). This is the same as the principle of reflection in optics. In some instances, light encounters a material that will reflect the rays back, which, at its core, is the key principle of how we see objects in the real world, as all of our vision depends on light reflecting off objects to reach our eyes (see chapter 1 ).
Laws of reflection
In simple terms, there are two laws of reflection to remember, and they apply to both plane and curved surfaces:
First law of reflection: The incident light ray and the reflected light ray lie in one plane.
This means that the reflected ray exists along the same line (and on the same side of the reflecting surface) as the incident (approaching) ray.
Second law of reflection: The angle of incidence (i) is equal to the angle of reflection (i′).
Angles of incidence and reflection are always measured relative to the ‘normal’ of the surface, with the normal being the hypothetical line that exists perpendicular to the surface at the point where light meets the surface. In Fig. 6.2 you can see that, relative to the normal, the angle of incidence (i) is identical to the angle of reflection (i ′ ). This second law of reflection also leads on to the reversibility principle, which means light would reflect back along the same path; in Fig. 6.2 this would mean if the reflected ray became the incident ray, the incident ray would become the reflected ray.
Image formation – plane surfaces
The easiest way to understand a plane mirror (flat reflective surface) is to find a mirror in your house that does not possess magnification or produce distortion. Usually a standard hallway/bathroom/full-length mirror will be a plane mirror, unless it’s designed to be curved. If you stand directly in front of a mirror like this, you will see an image of yourself which is the same size as you are in the real, physical world. Importantly, the image of yourself will appear to be standing directly in front of you – this is because the light leaving your body to meet the mirror is travelling along the normal of the mirror, so it travels back along exactly the same line. It is also interesting to note that the image of you in the mirror will be positioned as far within the mirror as you are standing in front of it, so if you’re 30 cm in front of the mirror, your alternate dimension mirror self will also be 30 cm within the mirror. Another way of saying this might be that the object distance (l) is equal to the image distance (l ′ ) with a plane mirror ( Fig. 6.3 ).
If we now consider a slightly more complex example, let’s say we have an object in front of our mirror, like shown in Fig. 6.3 . Now you can see that the incident light rays travelling along the normal (labelled 1) produce reflected rays that travel back along the same path. However if we stood in a different position without moving the object, then the image of the object would stay in the same place, but the incident and reflected rays from the object that we utilise to ‘see’ it will now be travelling along different paths. In Fig. 6.3 this is shown as the labelled rays 1 and 2.
Crucially, what we can also notice about the image of ourselves (or an object) in a plane mirror is that it will be laterally inversed (flipped from left to right), so if we raise our right hand, our mirror self appears to raise their left hand. Similarly the image will be reversed (flipped front to back), so we see ourself facing the front side of ourselves, indicating our image is facing the opposite way to us. Fig. 6.4 shows an example of this.
To summarise:
- 1.
The image is as far within the mirror as the object is in front.
- 2.
The imaginary line joining the object and the image is perpendicular to the mirror surface (irrespective of the position it is viewed from).
- 3.
The image is the same size as the object.
- 4.
The image is virtual, reversed and laterally inverted.
In summary, if we know the angle of incidence (i) of a ray of light approaching a fixed, plane mirror, then we can also easily determine the angle of reflection (i ′ ). So what if we complicate things a little by introducing a second plane mirror into the mix?
Multiple plane mirrors
Up to this point, we have discussed the angle of reflection (i ′ ) but not the angle of deviation (d) – defined as the angle between the original path of the ray and the actual reflected ray. Most of the time this angle isn’t necessary with reflection at plane surfaces, but when we have two mirrors it becomes incredibly relevant, so let’s start by doing some revision of angles with a single plane mirror. In Fig. 6.5 A, you can see that a ray of light is incident upon a plane mirror and is reflected at an angle identical to the angle of incidence (relative to the normal, shown as the orange dashed line). However, in this image, I’ve also drawn a hypothetical continuation of the incident ray (blue dashed line), into the mirror, to show what the ray would have done had the mirror not been obstructing its path. The angle of deviation (d), then, will be measured as the angle from the continuation line and the reflected ray. First though, let’s determine what we know about the angles – as you can see in Fig. 6.5 B, the total angle of deviation (d, shown in green) can be split into two smaller angles which are separated by the mirror. These angles will be equal to one another, and we know that because we can determine that the angle between the incident (blue solid) ray and the mirror will be equal to the angle between the reflected (purple solid) ray and the mirror. Then, we can use the law of opposite angles to determine that the angle between the continuation (blue dashed) ray and the mirror will be the same as the angle between the incident (blue solid) ray and the mirror (for revision on opposite angles, see chapter 2 , Box 2.4 ).
Great, but what has this got to do with two plane mirrors, you ask? Well, in optics, if we have two plane mirrors inclined at an angle towards one another, we can use these principles to determine the total angle of deviation (d), even though it has reflected off two mirrors ( Fig. 6.6 ). In this scenario, two plane mirrors are inclined towards one another at an angle (a) – importantly, this angle must be smaller than 180° because otherwise the light wouldn’t be able to reflect off both mirrors. If a ray of light is incident at one of the mirrors and reflects in such a way as to reflect off the second mirror, then the light will reflect back away from the mirrors, at an angle of deviation (d – relative to the original, incident ray). Importantly, both reflected rays (from mirror 1 and mirror 2) obey the laws of reflection (angle of incidence is equal to angle of reflection), and so we also know (thanks to Fig. 6.5 B) that the individual angles of deviation can be split into two identical angles. At the first mirror these angles are referred to as α (alpha), and at the second mirror, these angles are referred to as γ (gamma). If you’re good at visualising, you might also be able to see that the two α angles and the two γ angles would add together to produce the total angle of deviation. This means that we can calculate the angle of deviation using Equation 6.1 .
However, we probably don’t actually know the values of α and γ, which could hinder our ability to utilise this equation somewhat. Instead, let’s make use of the natural triangle formed by the angle between the mirrors (a) and the first reflected ray of light (shown in yellow in Fig. 6.7 ). In this triangle, we know that all internal angles need to add up to 180°, and we can also apply our knowledge of opposite angles (shown in Fig. 6.5 ) to determine that the previously unknown angle within the triangle must be equal to γ – but what does this tell us? Well, it tells us that:
