Total Internal Reflection, Critical Angle and Its Application

When the internet became popular back in the day, it took ages to send an email; true, the computers were not fast enough, but what truly changed the game was the advancements in the techniques involved in the data transfer. Since the adoption of optical fibres in telecommunications, data transfer speed has increased dramatically, and the data can be transferred much more efficiently over large distances. 

What if I told you that the diamond’s shine could be attributed to the same phenomenon that guides the data through an optical fibre, and this phenomenon is the total internal reflection? Optic fibres are basically light pipes, pipes that carry light. This is possible through a phenomenon known as total internal reflection. A mirage in a desert can also be attributed to this phenomenon. But total internal reflection does not happen under normal circumstances; it requires reflection to occur at an angle greater than the critical angle. Let us learn in detail about total internal reflection, critical angle, and its applications.

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What is Total Internal Reflection?

Refraction is the bending of light as it travels from one medium to another of different refractive indexes. When a ray of light moves from one transparent medium to another, its path depends upon the nature of the two mediums and the angle at which the ray strikes the interface between the two mediums.

When a ray of light moving from one medium to another undergoes complete reflection back into the medium that it came from, such a phenomenon is the total internal reflection of light. It occurs when a ray of light travels from a denser medium to a rarer medium. A ray of light that undergoes total internal reflection moves back into the same medium from which it came from, and since the light is completely reflected back, it is known as total internal reflection.

Understanding Total Internal Reflection (TIR)

Consider the situation given below:

A point source of light, marked as \(O\), is kept at the corner of a denser medium. Let this medium be water. \(XY\) represents the interface that separates the denser medium from the rarer medium (let it be air). As the light rays move from a denser medium to the rarer medium, it undergoes refraction and moves away from the normal. Suppose we increase the angle of incidence, the angle of refraction increases. At a certain value of angle of incidence, let it be \(i_c\), the angle of refraction becomes \(90^\circ \), and when that happens, the light ray coming from the source grazes at the interface separating the two mediums or, in simple words, the ray of light moves along the boundary \(XY\). But when the angle of incidence becomes greater than \({i_c}\), the angle of refraction cannot increase further; instead, the ray moves back into the denser medium. The ray is reflected, and this phenomenon is known as total internal reflection. The reflected ray obeys the laws of reflection. In such a situation, there is no refracted ray.

When rays of light strike a polished surface such as a mirror, some energy of the light rays are absorbed while some get transmitted. Thus, in mirrors, the intensity of the reflected rays is less than the intensity of the incident rays, and as a result, images formed are somewhat dull.  In total internal reflection, the intensity of the reflected ray is the same as the intensity of the incident ray. Hence the images formed by the total internal reflection of light are brighter.

Critical Angle

The angle of incidence above which total internal reflection of light takes place is known as the critical angle. When the angle of incidence is equal to the critical angle, the angle of refraction is equal to \(90^\circ \). It is represented by \({i_c}\). For the glass-air interface, the critical angle is maximum for red colour and minimum for violet colour.

Calculation of the Critical Angle

Let us suppose that a ray of light travels from a denser medium(1) into a rarer medium(2). Let \({\mu _1}\) and \({\mu _2}\) be the refractive indices of the denser and rarer medium, respectively. Let the ray of light be incident at an angle \(i\) and refracted at an angle \(r\)

From Snell’s law,

\(\frac{{{\mu _2}}}{{{\mu _1}}} = \frac{{\sin \,i}}{{\sin r}}\)

For critical angle, put \(i\, = \,c\) and \(r\, = \,{90^ \circ }\)

\(\frac{{{\mu _2}}}{{{\mu _1}}} = \frac{{\sin \,c}}{{\sin {{90}^ \circ }}}\)

\(\sin c = \frac{{{\mu _2}}}{{{\mu _1}}}\)

The critical angle for the two mediums 1 and 2 is thus given by:

\(c\, = \,\left( {\frac{{{\mu _1}}}{{{\mu _2}}}} \right)\)

If air is the rarer medium, substitute, \({\mu _2}\, = \,1\) we get, 

\(\sin c = \frac{1}{{{\mu _1}}}\)

\(c\, = \,\left( {\frac{1}{{{\mu _1}}}} \right)\)

If the angle of incidence is greater than the critical angle, the total internal reflection will occur.

Let us summarize the requirements for total internal reflection

(i) The light falling at the interface of the two transparent mediums should travel from a denser medium into a rarer medium
(ii) The angle of incidence should be greater than the critical angle for a ray of light to be reflected internally.

Applications of Total Internal Reflection

The phenomenon of total internal reflection is responsible for the functioning of various optical instruments like periscopes, binoculars, spectroscopes, and microscopes. Here are a few more applications of the total internal reflection:

1. Optical Fibre

An optical fibre consists of two layers; an outer layer of fibre is composed of a lower refractive index material, while the inner fibre is composed of a higher refractive index material. The inner layer is called the core, while the outer layer is called cladding. The light enters the fibre from one end of the core at an angle of incidence greater than the critical angle. It strikes the interface between denser to rarer medium, and it undergoes total internal reflection within the fibre multiple times before coming out from the other end.

Outside the cladding is a plastic jacket that protects the optic fibre from damage.

Optical fibres are used in:

a. Endoscopy
b. Optical communication

Optical fibres are used in communication

A modulating laser converts the electrical or radio signal into optical signals. These optical signals enter from one end of the fibre at an angle greater than the critical angle, undergo total internal reflection several times along the fibre, and get transmitted to the required location. At this end, the optical signals get converted back into electrical signals.

The loss in the intensity of the signal is less in optical fibres compared to the copper cables, and these fibres allow multiple signals to be transmitted simultaneously.

2. Diamonds

Total internal reflection is the phenomenon responsible for a diamond’s sparkling effect and is often used while polishing the diamonds. The refractive index of diamond is \({\mu _d}\, = \,2.8\) while the refractive index of air is \({\mu _d}\, = \,1\) For a ray of light travelling from diamond to air, the change in the refractive index is huge. Thus, the critical angle for light moving from diamond to air is quite small, just \({24.4^ \circ }\)

While the diamond polishing is done, specific cuts are made on the diamond surfaces to ensure light rays strike the inner surface at angles greater than the critical angle as much as possible. Thus light rays entering a diamond undergo multiple total internal reflections before coming out, thereby giving it a sparkling effect.

3. Mirage

A mirage is an optical illusion that deceives travellers moving in the desert during hot summer days. In a hot desert, the layer of air closest to the ground is hot and therefore rare, being in direct contact with the hot sands. Successive upper layers are cooler and denser. A ray of light moving down towards the earth keeps bending away from the normal as it travels into lower hotter layers. At a large enough angle of incidence \((i < c)\) the ray will undergo total internal reflection. The reflected ray will now keep bending towards the normal as it travels upwards into ever denser layers of air. This is why light is observed to have a curved path as the refractive index keeps increasing as you move up from the surface.  

If the light rays come from the blue sky, this reflection appears as a shimmering surface and is often misinterpreted as water by a thirsty man in a desert. If the light rays originate from a distant object on land, you will observe an inferior inverted image.

Mirages can be seen on roads; for a driver looking at the road at a low angle, the light from the objects over his head will follow a curved path. Thus, the observer receives two rays of light- one that follows a curved path and the other that follows a straight-line path, creating an illusion.

Solved Examples on Total Internal Reflection, Critical Angle, and its Applications

Q.1. Optical fibre has a refractive index of \({\mu _1}\, = 2\) covered by another material of refractive index \({\mu _2}\) Calculate the value of the refractive index of the material such that the critical angle between the two is \({60^ \circ }\)
Ans: We are given,\({\mu _1}\, = 2\)
\(c = {60^ \circ }\)
\(\sin c = \frac{{{\mu _1}}}{{{\mu _2}}}\)
\(\sin \,{60^ \circ } = \,\frac{{{\mu _2}}}{2}\)
Thus, \({\mu _2}\, = \,1.732\)

Q.2. Find the refractive index of a glass which has a critical angle of \({45^ \circ }\)
Ans: Critical angle, \(\theta  = {45^ \circ }\)
Refractive index \(\mu \, = \,\frac{1}{{\sin \,\theta }}\)
\(\mu \, = \,\frac{1}{{\sin \,{{45}^ \circ }}}\)
Thus, \(\mu \, = \,1.4\)

Summary

When a ray of light moving from one medium to another undergoes complete reflection back into the medium that it came from, such a phenomenon is known as the total internal reflection of light. The minimum angle of incidence for which total internal reflection of light takes place is the critical angle. When the angle of incidence is equal to the critical angle, the angle of refraction is equal to \(90^\circ \). Mathematically, the critical angle for a medium can be given as,\(c = \,\left( {\frac{1}{{{\mu _d}}}} \right)\) for light moving from the medium into air.

The phenomenon of total internal reflection is the principle behind the working of an optical fibre; it is responsible for the functioning of periscopes, binoculars, and microscopes and is the reason for the sparkling of diamonds, etc.

Frequently Asked Questions (FAQs)

Q.1. What are the basic conditions for total internal reflection?
Ans: Two conditions are:
1. The light must travel from a denser medium to a rarer medium.
2. The light rays must be incident at an angle greater than the critical angle.

Q.2. Give the formula for critical angle?
Ans: The critical angle for medium/air interface can be calculated using the formula: \(c = \,\left( {\frac{1}{{{\mu _d}}}} \right)\)

Q.3. Give a few examples of total internal reflection in everyday life.
Ans: A few examples are:
1. Sparkling effect in diamonds
2. Mirages
3. Periscopes

Q.4. What is the critical angle?
Ans: The minimum angle of incidence for which the angle of refraction equals \(90^\circ \) is called the critical angle.

Q.5. Define total internal reflection.
Ans: When a ray of light moving from one medium to another undergoes complete reflection back into the medium that it came from, such a phenomenon is known as the total internal reflection of light.

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