Jump to a key chapter

## Ray Tracing in Lenses and Mirrors

Drawing the paths of individual rays of light can be a useful way to visualize the behaviour of light interacting with lenses and mirrors. Using the **law of reflection** and **Snell's law** of refraction allows us to calculate the direction in which rays of light will travel and draw them in a **ray diagram**. As a reminder, the laws are the following.

### 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.

### 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.

$\frac{\mathrm{sin}\left({\theta}_{2}\right)}{\mathrm{sin}\left({\theta}_{1}\right)}=\frac{{v}_{2}}{{v}_{1}}=\frac{{n}_{1}}{{n}_{2}}$

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 the law of reflection**

The ray diagram below shows an incident ray traveling at$45\xb0$\(45^\circ\) towards a flat mirror positioned at a$10\xb0$\(10^\circ\) slope. Calculate the angle of the reflected ray.

As the mirror is positioned at an angle of \(10^\circ\) from the horizontal, we know that its normal (perpendicular to the surface) will be at an angle of \(10^\circ\) from the vertical. We can draw this line from the point that the ray strikes the mirror. The angle of incidence is the angle between the incident ray and the surface normal at the point the ray strikes. If the mirror was horizontal, the perpendicular axis would be vertical and the angle of incidence would be \(45^\circ\), due to alternate angles between parallel lines being equal.

However, as the perpendicular axis is at an angle of \(10^\circ\) from the vertical, the angle of incidence is:

\[q=45^\circ+10^\circ=55^\circ.\]

Therefore, we know that the angle of reflection \(q\) is also \(55^\circ\) due to the law of reflection. To work out the angle of the reflected beam with respect to the vertical/horizontal axes, we have to add the \(10^\circ\) slope of the axis perpendicular to the mirror to our calculated \(55^\circ\) angle of reflection:

\[55^\circ+10^\circ=65^\circ.\]

This means the reflected ray travels at an angle of \(65^\circ\) to the vertical, or \(25^\circ\) to the horizontal as shown below.

**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.

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

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.

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, \(p\):

$i$

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.

Curved mirrors are also called **spherical mirrors**, if their surface is the shape of a section of a sphere.

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.

Curved mirrors can also have a **parabolic** shape - this is a surface generated by rotating a parabola around its axis. Parabolic mirrors are less common than spherical ones, but they perform better - focussing an incoming collimated beam of light to a more precise focal point, with less spherical aberration.

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.

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.

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.

**Real images **appear when reflected rays intersect on the same side of the mirror as the object is, while **virtual images **form when the extended rays intersect behind the mirror on the opposite side.

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 \(f=\infty\).

## Ray Tracing Mirrors - Key takeaways

- Convex and concave mirrors are the shape of a section of a spherical surface, meaning their curvature can be defined using the radius \(r\). A Plane mirror (flat mirror) can also be considered as a spherical mirror with an infinite radius: \(r=\infty\).
- A mirror forms an image of an object at the point at which the rays originating from the object and reflected from the mirror
**intersect**. If the reflected rays**diverge**, we extend them backward to find an intersection point behind the mirror. If they**converge**, the rays will intersect at a point in front of the mirror. - In a plane mirror, the distance \(p\) between the mirror surface and object is equal to the distance \(i\) between the mirror and image. In curved mirrors, the distances \(p\) & \(i\) are not equal.
- If the reflected rays converge behind the mirror, the image formed is
**virtual**, meaning it cannot be projected onto a surface as the real rays never actually travel to the intersection point. If the reflected rays converge in front of the mirror they form a**real**image - this image could be projected onto a surface, as the real light rays actually pass through the intersection point. - Concave mirrors form a
**real focus**in front of the mirror surface at a focal length \(f=\frac{1}{2}r\). Convex mirrors form a**virtual focus**behind the mirror surface at \(f=\frac{1}{2}r\). - Convex and plane mirrors can only form a virtual image. Concave mirrors can form a
**real image**when the object is placed outside the focal point - the only scenario where a real image is formed - or a**virtual image**if it is placed inside the focal point. No image is formed when the object is exactly at the focal point of a concave mirror. - The object distance \(p\), image distance \(i\), focal length \(f\) and radius of curvature \(r\) are related by the
**mirror equation**: \(\frac{1}{p}+\frac{1}{i}=\frac{1}{f}=\frac{2}{r}\).

###### Learn with 8 Ray Tracing Mirrors flashcards in the free StudySmarter app

We have **14,000 flashcards** about Dynamic Landscapes.

Already have an account? Log in

##### Frequently Asked Questions about Ray Tracing Mirrors

How does a convex mirror form an image?

Light rays originating from an object diverge as they approach the mirror. The outwardly-curved surface of a convex mirror means the reflected rays **diverge more** than the incident rays. This means to find a point where the reflections intersect, we have to extend the rays behind the mirror - forming a **virtual image. **

How does a concave mirror form an image?

Light rays originating from an object diverge as they approach the mirror. The inwardly-curved surface of a convex mirror means the reflected rays **converge**** more** than the incident rays. This means that the light can converge at various points and form a real or virtual image depending on where the object is placed.

What is a ray diagram?

A ray diagram is a diagram that traces the path of light rays to understand how they travel through a system of refracting and reflecting objects. Ray diagrams can be used to plot the paths taken by light rays for an observer to view the image of an object reflected in a mirror, and to help us to understand the properties of the image formed.

What process forms an image in a mirror?

When we view a point on an object, our visual systems interpret the light rays detected by our eyes by relying on the fact that the rays come from the point they originated. As the rays reflected by a mirror do not originate from a single point on the mirror, our brains interpret this by extending the rays **behind **the mirror to find their intersection point, which is where we perceive the image to appear.

Can a convex mirror form a real image?

**No** - the rays reflecting from a convex mirror will always be diverging, meaning they cannot have an intersection point in front of the mirror surface. The only type of mirror that forms a real image is a concave mirror, when the object is positioned outside the focal length.

##### About StudySmarter

StudySmarter is a globally recognized educational technology company, offering a holistic learning platform designed for students of all ages and educational levels. Our platform provides learning support for a wide range of subjects, including STEM, Social Sciences, and Languages and also helps students to successfully master various tests and exams worldwide, such as GCSE, A Level, SAT, ACT, Abitur, and more. We offer an extensive library of learning materials, including interactive flashcards, comprehensive textbook solutions, and detailed explanations. The cutting-edge technology and tools we provide help students create their own learning materials. StudySmarter’s content is not only expert-verified but also regularly updated to ensure accuracy and relevance.

Learn more