Total internal reflection

Total internal reflection (TIR) happens when light travelling inside a material such as glass or water hits the boundary with a material of lower refractive index, such as air, at an angle bigger than the critical angle. None of the light escapes: all of it reflects back inside. Optical fibres, the prisms in binoculars and the sparkle of a diamond all depend on it.

What is total internal reflection?

When light crosses a boundary between two transparent materials, it changes speed and bends: it refracts. Going from glass into air, it speeds up and bends away from the normal (the line at right angles to the surface). Usually a little light is also reflected back. As the angle of incidence increases, the refracted ray bends further from the normal, until at one angle it would have to leave at 90°, skimming along the surface. Beyond that angle, light can't refract out at all, so all of it is reflected.

Total internal reflection needs both of these conditions:

1. The light is travelling from a material with a higher refractive index into one with a lower refractive index, for example from glass into air or from water into air.
2. The angle of incidence is greater than the critical angle for that pair of materials.

It can never happen the other way round. Going from air into glass, light slows down and bends closer to the normal, and some of it always gets through.

What is the critical angle?

The critical angle, C, is the angle of incidence at which the angle of refraction is exactly 90°. Below it, light refracts out (with a weak reflection). At it, the refracted ray runs along the boundary. Above it, there is total internal reflection, and the ray bounces back at the same angle to the normal as it arrived.

Every angle in the figure is drawn true: in the first case, Snell's law gives an angle of refraction of 48.6° for light leaving glass at 30°.

Critical angle formula

Put an angle of refraction of 90° into Snell's law, n₁ sin θ₁ = n₂ sin θ₂. Since sin 90° = 1, the critical angle for light going from material 1 to material 2 is given by:

When the second material is air, its refractive index is about 1, so this becomes:

The bigger the refractive index, the smaller the critical angle, and the easier it is to get total internal reflection.

Worked examples

Example 1: critical angle of glass

Find the critical angle for light going from glass of refractive index 1.50 into air.

1. sin C = 1 ÷ 1.50 = 0.667
2. C = sin⁻¹(0.667) = 41.8°

Example 2: refractive index from the critical angle

In an experiment with a semicircular block, the refracted ray first runs along the flat face when the angle of incidence is 39.0°. What is the refractive index of the block?

1. At the critical angle, sin C = 1/n, so n = 1 ÷ sin C.
2. n = 1 ÷ sin 39.0° = 1 ÷ 0.629 = 1.59

Example 3: will it reflect?

A ray inside a diamond meets a face at an angle of incidence of 30°. Does it escape? The critical angle for diamond is 24.4°, and 30° is bigger, so the ray is totally internally reflected. Light can bounce around inside a cut diamond many times before it finds a way out, which is why diamonds sparkle.

Example 4: between two materials (A-level)

An optical fibre has a core of refractive index 1.62 and cladding of refractive index 1.52. Find the critical angle at the core to cladding boundary.

1. sin C = n₂ ÷ n₁ = 1.52 ÷ 1.62 = 0.938
2. C = sin⁻¹(0.938) = 69.8°

That is far bigger than the 41.8° for glass to air, since 1.52 and 1.62 are close together. Light must hit the boundary at a grazing angle, more than 69.8° from the normal, to stay in the core. Check any of these in the refraction calculator.

Now try one: decide what a ray does at a boundary, then press Mark to check your answer in the Light and Energy course.

Optical fibres

An optical fibre is a thin strand of glass or plastic that carries light along it by total internal reflection. Light enters one end and hits the side of the fibre at an angle greater than the critical angle, so it reflects. It keeps reflecting all the way along, even round gentle bends, and comes out of the other end.

• The core carries the light.
• The cladding surrounds the core and has a lower refractive index, so total internal reflection happens at the core to cladding boundary. It also protects the core's surface from scratches, and in a bundle it stops light crossing from one fibre into the next.

Optical fibres carry internet and telephone signals as pulses of light, and bundles of them are used in endoscopes, which let doctors see inside the body: one bundle carries light in and another carries the image out.

Pulse broadening (A-level)

AQA's A-level specification asks for step index fibres, material and modal dispersion, and the effects of pulse broadening and absorption.

• Modal dispersion: rays entering at different angles take different paths. Rays that zigzag more travel further and arrive later, so a pulse spreads out. A very narrow core, which allows only one path, reduces it.
• Material dispersion: the refractive index of glass depends slightly on wavelength, so different colours travel at different speeds and arrive at different times. Using light of a single wavelength (monochromatic light) reduces it.
• Absorption: the glass absorbs some of the energy, so the signal's amplitude falls along the fibre.

If pulses spread out too much, neighbouring pulses overlap and the signal can't be read, which limits how fast data can be sent.

Other uses of total internal reflection

• Prisms in binoculars and periscopes. Light hits the inside face of a 45° glass prism at 45°, which is more than glass's critical angle of about 42°, so it reflects completely. Unlike a silvered mirror, it reflects all of the light and has no coating to tarnish.
• Diamonds. Their very small critical angle traps light inside, so it reflects many times before leaving.
• Corner reflectors. Prisms with three faces at right angles send light straight back the way it came, and they reflect all of it when the conditions for total internal reflection are met.

Common mistakes

• Getting the direction wrong. TIR only happens going from the higher refractive index material to the lower one, never from air into glass.
• Measuring from the surface. Angles of incidence and the critical angle are measured from the normal, not from the boundary.
• Saying "greater than or equal to". At exactly the critical angle the light runs along the surface. TIR needs an angle bigger than C.
• Writing sin C = n. It is sin C = 1/n. A sine can't be more than 1, and refractive indices are bigger than 1.
• Calculator in radians. Make sure it is set to degrees before using sin⁻¹.

To practise refraction and other wave questions, try today's GCSE Physics Higher or A-level Physics: Year 1 short timed paper.

Read next

Physics · Waves and lightCritical angle formula: sin θc = n₂/n₁, with examplesThe critical angle formula sin θc = n₂ ÷ n₁ (or 1 ÷ n into air): how it is derived, total internal reflection, optical fibres, typical values and examples.