Dispersion of Light

The world around us is made up of varying shades of different colors. As a result, the perception of color of surrounding objects is subjective and differs for each of us. That being said, regardless of our subjective perception, the color of the light itself has an objective cause. It is determined by the oscillation frequency of the electromagnetic waves. Each color of light has its own characteristic range of frequencies. These colors can be obtained by directing a narrow beam of light at a prism. An alternative is going outside after a rainstorm in the hopes that you might see a rainbow.

Fig. 1 - A rainbow, forms as a result of light from the sun scattered by rain droplets due to dispersion.

As the sunlight gets refracted and reflected in the water droplets, it spreads out into the familiar order of seven colors. In this article, we will study the concept of the dispersion of light, and understand its causes and applications.

Dispersion of Light Definition

First, let's define what exactly is dispersion.

There are three different types of dispersion: modal dispersion, chromatic dispersion, and material dispersion. Before we explain each of these types, we must become familiar with optical fibers.

Light rays can travel in optical fiber in two different ways: in the same direction with the same path lengths, or in different directions with varying path lengths. Keeping that in mind, we can further explain the three types of dispersion mentioned earlier:

1. Modal dispersion occurs when light rays in an optical fiber travel in different directions, having additional path lengths. As a result, light rays reach the other end of the fiber at different times.

2. Chromatic dispersion occurs when there is a change in the propagation velocity of the rays with the wavelength in the optical fiber.

3. Material dispersion causes white light to split into different colors, depending on the material's refractive index.

In this article, we'll focus on material dispersion, one of the main examples being the splitting of light using a prism. The first person to recognize this property of light was Isaac Newton.

Dispersion of Light Experiment

Dispersion of light was first discovered in the 17^th century when Isaac Newton passed a beam of light through a tiny slit.

Fig. 2 - Isaac Newton performing his famous light experiment, separating the white light into its components.

As he directed the light at the prism, he expected an identical beam of white light to come out on the other side. However, once the light left the prism, it formed a broad, colored band on the screen, arranged just like a rainbow, formed by the sunlight or white light. The seven primary colors of the rainbow he observed were red, orange, yellow, green, indigo, blue, and violet.

The color observed depended on the width of the slit. So through trial and error, it became clear that the violet rays of light bend the most so have the shortest wavelength of all the colors, while red bends the least corresponding to the longest wavelength.

Causes of Dispersion of Light

So, what exactly causes the dispersion of light? As we already established, light consists of several colors, which travel at different velocities in different mediums. The refractive index of a medium is the ratio of the velocity of light in a vacuum to the velocity of light in any medium and can be calculated using

\[n=\frac{c}{v},\]

where \(n\) is the index of refraction, \(c\) is the speed of light, and \(v\) is the velocity in a substance.

The speed of light in a vacuum is a fundamental constant of nature equal to \(299\,792\,458\,\frac{\mathrm{m}}{\mathrm{s}}\). It's slower than that in other transparent mediums, such as water, glass, etc. Therefore, due to changes in wave velocities of light in a medium, deviation or bending of light occurs due to different refractive indices.

Since different colors or wavelengths of light travel at different speeds in a particular medium, some colors bend more. Just as Newton observed, red light undergoes the least deviation and violet the most. Hence, dispersion allows us to see the colors individually. The most common method for observing dispersion is shining a beam of light through a prism.

Dispersion of Light Through A Prism

A prism is a tool that divides white light into a spectrum.

The angle at which the triangular surfaces are inclined is called the angle of the prism.

The bending of the light rays in the prism is determined by the law of refraction, also known as Snell's law. Mathematically, it can be expressed as

\[ n_1 \sin \theta_1 = n_2 \sin \theta_2,\]

where \(n\) is the index of refraction and \(\theta\) is the angle of incidence or refraction.

Fig. 3 - Dispersion of light by a glass prism, resulting in a band of different colors.

In the case of the glass prism, the ratio of the sines of the angles of incidence and refraction is equal to the ratio of the refraction coefficients in glass (\(1.52\)) and air \((1.0003\)).

Therefore, the explanation for the fact that the light rays corresponding to different colors bend at different angles in the prism is the result of the fact that the light refraction coefficient \(n\) depends on the frequency or the wavelength of the light.

In the visible light spectrum, red light has the smallest angle of inclination in the prism, and violet has the largest angle. A greater bending of a light beam in a medium means a greater refractive index. So, in a glass prism, the refraction coefficient increases as the frequency increases (or as the wavelength decreases). This also happens to other substances as long as they are transparent to light.

Dispersion of Light Examples

Let's look at some real-life examples of dispersion!

Rainbow

A rainbow is formed when the rays of the Sun's white light refract into water droplets in the atmosphere after rainfall. The incident light ray refracts in the droplet, reflects from the opposite side of the droplet, and refracts again when leaving it into the air. To observe a rainbow, the Sun must be shining behind the observer after the rain. The reflected sunlight enters the eyes from all those raindrops that reflect the light at an angle of \(42^\circ\). If the water drops are higher or lower, then the observer does not see the reflected rays at all.

Due to light dispersion, the outer band of the rainbow is red, and the inner band is violet. The refractive index of light in water for red light is around \(1.33\), and for violet, it is slightly higher, equal to \(1.34\). Therefore, the angle of reflection for red light is slightly above the incident angle of \(42^\circ \, 22'\), and for violet, it is smaller, only \(40^\circ \, 36'\).

The visible shades of light in the rainbow are also determined by the size of the raindrops. If they are \(1 \, \mathrm{mm}\) to \(2 \, \mathrm{mm}\) in diameter, then a bright violet, green and red circle is usually seen. As the droplet diameter decreases, the red color gradually becomes dimmer. When the droplets are only a tenth of a millimeter in diameter, only a blurred violet color is seen in the arc of the rainbow. If the drops are even smaller, then the rainbow is barely visible.

Spectrographs

Light dispersion is used in optical instruments to divide the incident light into a spectrum of colored bands or lines. According to the position of the bands or lines on the scale of the instrument, it is possible to determine the frequency or the wavelength of the incident light wave. Such devices are called spectrographs. The main components of these spectrographs are a slit, a self-elevating lens, and a prism. The task of the prism is to separate the rays of different lights; therefore, they are made of materials that have as much dispersion as possible in the required range of wavelengths. Spectrographs can be used to very accurately recognize the chemical composition of any material, including many celestial objects.