Ray Tracing Mirrors

Law of reflection

For a light ray bouncing off a surface, the angle of incidence is equal to the angle of reflection, measured from an axis perpendicular to the surface at the point the ray strikes. Both are represented by \(q\) in the diagram below.

The law of reflection states that the angle of incidence is equal to the angle of reflection, StudySmarter Originals.

Snell's law

Snell's law describes the relationship between the angle of incidence and the angle of refraction at a boundary between materials with different refractive indexes. The ratio between the sines of the angles of incidence and refraction is equal to the ratio of the speeds of light in each material, and to the reciprocal of the ratio of refractive indexes of the materials.

$\frac{\sin \left({\theta }_2\right)}{\sin \left({\theta }_1\right)}=\frac{v_2}{v_1}=\frac{n_1}{n_2}$

Using these two laws we can predict the path a ray of light will travel along through a system of lenses and/or mirrors using a ray diagram.

Ray Tracing Examples

Ray diagrams using Snell's law

A ray of light travels through air (\(n_1=1\)) towards a glass prism (\(n_2=1.5\)) as shown below. Determine the angles of the ray as it passes through the glass and after exiting the prism.

Snell's law example problem - a light ray approaches a glass prism at an angle 30 degrees from the surface normal. StudySmarter Originals.

The angle of incidence \(\theta_1=30^\circ\), so we can use Snell's law to calculate the angle of refraction \(\theta_2\) at the first boundary:

\begin{align*}&\frac{\sin\left(\theta_2\right)}{\sin\left(\theta_1\right)}=\frac{1}{1.5},\\&\theta_2=\arcsin\left(\frac{1}{1.5}\times\sin\left(30^\circ\right)\right)=19.5^\circ.\end{align*}

This enables us to calculate the angle the ray travels through the prism. The ray approaches the block horizontally (\(0^\circ\)) and has an angle of refraction of \(19.5^\circ\) at the first boundary. As the boundary is at a \(30^\circ\) slope, the angle the ray travels through the prism is \(10.5^\circ\) below horizontal:

\[30^\circ-19.5^\circ=10.5^\circ.\]

By drawing the path of the ray, we find that the next boundary it strikes is the right face of the prism, which is vertical. This means that the angle of incidence \(\theta_1=10.5^\circ\). We can then calculate the angle of refraction at this boundary:

\[\theta_2=\arcsin\left(\frac{1.5}{1}\times\sin\left(10.5^\circ\right)\right)=15.9^\circ.\]

As the second boundary that the ray travels through is vertical, the ray exits the prism traveling at an angle \(15.9^\circ\) below horizontal

Example solution for Snell's law, StudySmarter Originals.

The law of reflection and Snell's law can be used together to calculate the paths of rays through more complex systems involving both reflection and refraction. The same techniques can also be applied in three dimensions using 3D angles, but in this article, we will stick to 2D problems for simplicity.

Ray Tracing Mirror Reflections

When looking at an object in a plane (flat) mirror, the object appears to be positioned somewhere behind the mirror - but how can this be, since we can see the mirror is a thin object and there is no physical space behind it? The answer is that the image you see in the mirror is virtual - not existing in a real physical space.

Virtual images

When working out what image will be seen in a reflection, we can use ray tracing to draw the paths each light ray takes to arrive at the eye/camera. While in reality light travels from the object toward the observer, we can trace the paths of light rays originating from the observer to determine what objects they arrive at. As light travels in straight lines, these paths are the same for the light traveling in either direction between the object and observer.

In the diagram below, an observer views an arrow (the object) reflected in a plane mirror. We can trace the paths of light rays originating from the observer, reflecting off the mirror, and arriving at a point on the real object. These represent the real paths that light rays originating from the object travel to reach the observer.

Ray diagram showing how when an observer views an object in a mirror, the light rays appear to be converging from an object behind the mirror, StudySmarter Originals

When light rays reach the observer, they appear to be coming from behind the mirror. This is because the image that an observer perceives from the light reaching their eye is interpreted based on the fact that light only travels in straight lines. This means that the reflected rays are perceived to extend through the mirror surface, giving the appearance of originating from behind it and creating a virtual mirror image of the object. The virtual image appears to be at a displacement \(i\) behind the mirror, which is equal to minus the real perpendicular displacement between the object and the mirror, \(p\):

$i=-p$

for plane mirrors.

Ray Tracing in Concave and Convex Mirrors

As you might expect, the behavior of reflections becomes more complex when we introduce curved concave and convex mirrors. However, the same principles still apply and we can use ray tracing to understand what is going on. We will limit this explanation to convex and concave mirrors that are the shape of a small section of a spherical surface, so their curvature can be defined by the radius \(r\) of the sphere. In a concave mirror \(r\) is positive, in a convex mirror it is negative, and in a plane (flat) mirror \(r\) is infinite. The center of curvature \(C\) is the center point of the imaginary sphere with radius \(r\). The central axis is an axis between the center of the mirror surface and the center of curvature.

Concave and convex mirrors have a circular cross-section with radius \(r\). The central axis lies between the center of curvature \(C\) and the center of the mirror surface, StudySmarter Originals.

As we found earlier, for a plane mirror the magnitude of the image distance \(i\) is always equal to the object distance \(p\). Let's explore if this is the case for curved mirrors. The diagram below shows an object positioned in front of a concave and convex mirror. If we trace the paths of light rays emitted from a point on the object to the mirror and extend their reflections behind the mirror, we find that the image distance and object distance are no longer equal.

Extending the rays reflected from an object positioned next to a mirror to find their convergence point shows us where the mirror image will appear. In a concave mirror, the object is larger and further behind the mirror, while in a convex mirror it is smaller and closer, StudySmarter Originals.

A convex mirror creates a smaller image that appears closer (\(|i|<|p|\)). This means the field of view is larger in a convex mirror, as objects appear smaller so more can be viewed than in a concave mirror of the same size.

We can find a way to relate the radius of curvature to the object and image distance, but first, we need to establish the focal point of curved mirrors. Consider an object positioned on the central axis of a curved mirror, a large distance from the mirror surface. As the object is far away, we can consider the light rays it emits to be parallel when they reach the mirror, as shown in the diagram below.

In a concave mirror, the focal point is in front of the mirror, making it real. In a convex mirror, it lies behind the mirror, so it is a virtual focal point, StudySmarter Originals

By tracing the paths of the reflected rays, we find that they intersect at a point on the central axis - for a concave mirror, this is a real focus in front of the mirror, while for a convex mirror, it is a virtual focus behind the mirror. As the rays reflected in the concave mirror are the real paths the light travels, an image of the distant object would be projected onto a surface positioned at the focal point. In contrast, if we put a surface at the virtual focus of the convex mirror, no image is projected as the real rays never actually travel to this point.

For both types of curved mirrors, the focus point is positioned on the central axis at the focal length \(f\) given by:

\[f=\frac{1}{2}r.\]

Much like object and image distance, it is positive if it is in front of the mirror and it is negative if it is behind the mirror.

Ray Tracing Images in Spherical Mirrors

Placing an object at different positions around the focal point of a curved mirror causes the image to behave in different ways, as shown in the ray diagrams below.

The behavior of the image created by a concave mirror depends on the location of the object in relation to the focal point. [Left] If the object is between the focal point and the mirror, the mirror creates a virtual image. [Center] If the object is at the focal point, the reflected rays are parallel and extend to infinity without converging and forming an image. [Right] If the object is on the outside of the focal point, the reflection forms a real image positioned outside the focal point of the mirror, StudySmarter Originals.

• When the object is placed between the focal point and the mirror, an observer will see a virtual image of the object appearing to be behind the mirror.

• As the object is moved further away from the mirror the image appears to move further behind the mirror, until the object is positioned at the focal point. This produces a reflection with parallel rays, making the image appear at an infinite distance both in front of and behind the mirror. However, this image is impossible to observe since the reflected rays never converge in either direction.

• If the object is moved outside the focal point, the reflected rays now converge in front of the mirror to produce an inverted real image. If we placed a surface at the image position, the image would be projected onto it since the real paths the rays travel converge at this point. This is the only scenario in which a mirror can produce a real image.

The relationship between object distance \(p\), image distance \(i\), and focal length \(f\) is the mirror equation:

\[\frac{1}{p}+\frac{1}{i}=\frac{1}{f}=\frac{2}{r}.\]

This equation is true for any concave, convex, or plane mirror. For a plane mirror the curvature \(r=\infty\), which also means \(f=\infty\).