3. Refraction of Light

Refraction is the bending of light as it passes obliquely from one transparent medium into another of different optical density. This bending happens because light changes its speed on entering a new medium: it slows down while entering a denser medium and speeds up while entering a rarer (less dense) medium. Refraction is responsible for many everyday illusions, including why a pencil dipped in water appears bent and why a swimming pool looks shallower than it actually is.

3.1 Laws of Refraction (Snell's Law)

1. The incident ray, the refracted ray, and the normal at the point of incidence all lie in the same plane.
2. The ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant for a given pair of media and for a given colour (wavelength) of light. This is known as Snell's Law, named after the Dutch astronomer Willebrord Snell:

sin i / sin r = constant = n₂₁ (refractive index of medium 2 with respect to medium 1)

When light travels from a rarer medium (like air) to a denser medium (like glass or water), it bends towards the normal, and the angle of refraction is smaller than the angle of incidence. When light travels from a denser medium to a rarer medium, it bends away from the normal.

3.2 Refractive Index

The refractive index (n or μ) of a medium is a measure of how much that medium slows down and bends light compared to another reference medium (usually vacuum or air). The absolute refractive index of a medium is defined as:

n = speed of light in vacuum (c) / speed of light in the medium (v)

Since light always travels slower in a medium than in vacuum, the refractive index of any transparent medium is always greater than 1. A higher refractive index means the medium is 'optically denser' and bends light more. Refractive index has no units, as it is a ratio of two speeds (or, equivalently, a ratio of two sines).

3.3 Refraction Through a Glass Slab

When a ray of light passes through a rectangular glass slab with parallel faces, it bends towards the normal on entering the slab (going from rarer air to denser glass) and bends away from the normal by an equal amount on emerging from the slab (going from denser glass back to rarer air). As a result, the emergent ray is parallel to the original incident ray, but it is laterally displaced — shifted sideways from the path the original ray would have followed had it travelled straight through. This lateral displacement is why a pencil held behind a thick glass slab appears to be shifted, and it is also why objects viewed through water or glass appear closer to the surface than they really are (apparent depth is less than real depth), which is precisely why a swimming pool always looks shallower than it is, and why a coin at the bottom of a bucket of water appears to be raised up.

A closely related everyday observation is that a swimming pool or a pond always looks shallower than it actually is when viewed from directly above the water surface — this is because light rays from the bottom of the pool bend away from the normal as they emerge from water into air, so the apparent depth (as perceived by the eye, tracing the emergent rays backward) is less than the real depth. In fact, for near-vertical viewing, the ratio of real depth to apparent depth is approximately equal to the refractive index of the medium, which for water gives real depth ≈ 1.33 × apparent depth — a relationship occasionally used in simple numerical questions.

3.4 Total Internal Reflection (TIR)

When light travels from a denser medium into a rarer medium, it bends away from the normal. As the angle of incidence in the denser medium is increased, the angle of refraction also increases and gets closer to 90°. At one particular angle of incidence, called the critical angle (C), the angle of refraction becomes exactly 90°, meaning the refracted ray grazes along the boundary of the two media. If the angle of incidence is increased beyond this critical angle, the light is no longer refracted at all — instead, it is completely reflected back into the denser medium. This phenomenon is called total internal reflection.

Total internal reflection therefore occurs only when two conditions are satisfied simultaneously:

1. Light must be travelling from a denser medium to a rarer medium (for example, from water or glass into air).
2. The angle of incidence in the denser medium must be greater than the critical angle for that pair of media.

The critical angle depends on the refractive index of the medium: a medium with a higher refractive index has a smaller critical angle. For the water-air boundary the critical angle is about 48–49°, and for the glass-air boundary it is about 42°, while for diamond (very high refractive index) it is only about 24°, which is unusually small. Mathematically, the critical angle C and the refractive index n of the denser medium (with respect to the rarer medium) are related by sin C = 1/n, which follows directly from applying Snell's law at the boundary with the angle of refraction set to exactly 90°.

It is worth clearly distinguishing total internal reflection from ordinary reflection. In everyday (partial) reflection off a mirror or any polished surface, only a fraction of the incident light is reflected while some may be absorbed or transmitted; but in total internal reflection, as the name suggests, 100 percent of the light incident on the boundary is reflected back into the denser medium with no loss at all, which is precisely why devices relying on TIR (such as optical fibres and prism binoculars) can transmit or redirect light so efficiently over long distances or through many internal bounces without becoming noticeably dim.

Applications of total internal reflection

Optical fibres: An optical fibre is a thin, flexible strand of extremely pure glass or plastic. Light entering one end strikes the internal walls of the fibre at an angle greater than the critical angle every time, and so is totally internally reflected again and again along the length of the fibre, travelling without significant loss even around bends, until it emerges at the other end. Optical fibres are widely used in telecommunications (carrying telephone and internet signals over long distances with very little loss), and in medicine in devices called endoscopes, which allow doctors to see inside the human body.

Mirage: A mirage is an optical illusion commonly seen on hot roads or in deserts, where a shimmering pool of water appears to be present at a distance, especially on hot summer days. It occurs because air near a very hot surface (such as a road) becomes much less dense (hotter) than the air above it, creating layers of air with gradually changing refractive index. Light from the sky, while travelling through these layers towards the observer's eye, bends gradually and eventually undergoes total internal reflection at the layer closest to the ground, and reaches the eye as though it were coming from a reflecting water surface — the observer perceives an inverted image of the sky, which looks exactly like a pool of water.

Sparkle of a diamond: Diamonds have an unusually high refractive index (about 2.42) and, correspondingly, a very small critical angle (about 24°). Gem-cutters exploit this by cutting the facets of a diamond at precise angles so that light entering the diamond undergoes multiple total internal reflections inside the stone before emerging, giving the diamond its brilliant sparkle.

Prism binoculars and periscopes: Right-angled prisms (with a 45°-45°-90° cross-section) are used instead of ordinary mirrors in binoculars, periscopes, and single-lens reflex (SLR) cameras to turn light through 90° or 180° by total internal reflection. Prisms are preferred over silvered mirrors in precision optical instruments because total internal reflection reflects almost 100% of the incident light with no loss due to absorption by a silvering layer, giving a brighter and clearer image.