Refraction of light

Cambridge·Physics·3.2.2·Extended

Refraction at a boundary

Light travels in a straight line through a uniform transparent medium. When it crosses into a different transparent medium, its speed changes. This can also change the light’s direction. The change in direction at the boundary is called refraction.

For a ray that meets the boundary at an angle:

  • If light travels more slowly in the new medium, the ray bends towards the normal.
  • If light travels faster in the new medium, the ray bends away from the normal.

For example, when light passes from air into glass, it slows down and bends towards the normal. When it passes from glass into air, it speeds up and bends away from the normal.

The normal and the angles

The normal is an imaginary line perpendicular to the boundary at the point where the ray meets it. It provides the reference line for measuring both angles:

  • The angle of incidence is between the incoming, or incident, ray and the normal.
  • The angle of refraction is between the refracted ray and the normal.

Both angles are measured from the normal, not from the boundary. Bending towards the normal means the angle of refraction is smaller than the angle of incidence; bending away means it is larger.

A ray along the normal

A ray that meets the boundary along the normal has an angle of incidence of zero. Its speed changes as it enters the new medium, but its direction does not change. It continues straight along the normal.

Tracing refraction through transparent blocks

A narrow ray of light can change direction when it crosses between air and a transparent block. This change of direction is called refraction. Tracing the ray shows where it enters and leaves the block and reveals its path inside.

Tracing the ray

  1. Place the transparent block on a sheet of paper and draw around its outline. Leave the block in place.
  2. Direct a narrow ray of light at one surface of the block. Mark points along the incident ray approaching the block and the emerging ray leaving it.
  3. Mark the points where the ray enters and leaves the block. Then remove the block.
  4. Use a ruler to draw the incident and emerging rays through their marked points. Join the entry and exit points to show the ray’s path inside the block.

Measuring the angles

At each point where the ray crosses a boundary, draw a normal: a line perpendicular to the surface at that point. Use a protractor to measure the relevant angles between the rays and the normal, not between the rays and the surface.

At the entry point, measure the angle of the incident ray and the angle of the ray inside the block. At the exit point, measure the angle of the ray inside the block and the angle of the emerging ray.

Interpreting the traced path

The ray travels in a straight line within a uniform block. Its direction may change at each boundary where it crosses between air and the block; a ray directed along the normal does not change direction at that boundary.

Repeat the trace with different incident directions and differently shaped blocks. The orientation of each boundary determines the direction of its normal, so it affects the angles at which the ray meets and leaves that boundary. Comparing the traces shows how the boundary orientations affect the ray’s passage through the block.

Total internal reflection

When light reaches a boundary from a material in which it travels more slowly, some of it may reflect back into that material while the rest crosses the boundary and changes direction. This change of direction is called refraction.

If the material on the other side is one in which light travels faster, the refracted ray bends away from the normal. The normal is an imaginary line perpendicular to the boundary. Angles of incidence, reflection and refraction are all measured from the normal.

Critical angle and total internal reflection

As the angle of incidence in the slower medium increases, the angle of refraction increases. At one particular angle of incidence, the refracted ray travels along the boundary, at an angle of refraction of $$90^\circ$$. This angle of incidence is the critical angle.

For an angle of incidence greater than the critical angle, all the light reaching the boundary is reflected back into the first medium. No refracted ray crosses the boundary. This is total internal reflection. The angle of reflection equals the angle of incidence.

Total internal reflection requires both of these conditions:

  • Light must approach the boundary from the slower medium towards the faster medium.
  • The angle of incidence must be greater than the critical angle.

Investigating total internal reflection with a semicircular block

Direct a narrow ray through the curved surface of a transparent semicircular block towards the centre of its flat face. The ray meets the curved surface normally, so it enters without changing direction. It then reaches the flat face from inside the block.

Vary the angle at which the ray meets the flat face and observe the internally reflected and refracted rays. As the angle of incidence increases, the refracted ray moves closer to the flat boundary. At the critical angle, it travels along that boundary. Beyond the critical angle, no refracted ray emerges through the flat face: the light is totally internally reflected.

Uses of total internal reflection

In an optical fibre, repeated total internal reflections guide light along the fibre. In binoculars, prisms use total internal reflection to redirect light.

Refractive index

When light passes from one region into another, its speed can change. The refractive index of region 2 relative to region 1 compares its speed in the two regions:

n=v1v2n = \frac{v_1}{v_2}

Here, $$v_1$$ is the speed of light in region 1 and $$v_2$$ is its speed in region 2. A larger value of $$n$$ means that light travels more slowly in region 2 relative to region 1. If $$n > 1$$, light is slower in region 2; if $$n < 1$$, it is faster there.

Refractive index and angles

The change in speed is related to the change in the light’s direction at the boundary. The angle of incidence $$i$$ and angle of refraction $$r$$ are both measured from the normal, an imaginary line perpendicular to the boundary. For light travelling from region 1 into region 2:

n=sin⁡isin⁡rn = \frac{\sin i}{\sin r}

When $$n > 1$$, light slows down on entering region 2 and bends towards the normal, so $$r < i$$. When $$n < 1$$, it speeds up and bends away from the normal, so $$r > i$$.

The relationship can be rearranged to find either angle, provided the light enters region 2:

r=sin⁡−1(sin⁡in),i=sin⁡−1(nsin⁡r)r = \sin^{-1}\left(\frac{\sin i}{n}\right), \qquad i = \sin^{-1}(n\sin r)

Refractive index and critical angle

For light travelling inside a material towards a boundary with air, the critical angle $$c$$ is the angle of incidence that produces a refracted ray along the boundary. It is measured from the normal inside the material.

For a material with a higher refractive index than air, its refractive index $$n$$ relative to air is approximately:

n=1sin⁡cn = \frac{1}{\sin c}

The same relationship can be rearranged to calculate the critical angle from the refractive index:

c=sin⁡−1(1n)c = \sin^{-1}\left(\frac{1}{n}\right)

Optical fibre communication

An optical fibre carries information as pulses of light through a transparent core. The core is surrounded by cladding, which has a lower refractive index than the core.

Keeping light inside the fibre

As light travels along the core, it meets the boundary between the core and the cladding. If its angle of incidence is greater than the critical angle, the light undergoes total internal reflection: it reflects back into the core rather than passing into the cladding. The angle of incidence is measured from the normal, an imaginary line perpendicular to the boundary.

The light can undergo repeated total internal reflections as it travels, keeping it within the core along the fibre.

Sending and receiving information

In telecommunications, information is encoded in light signals and sent through optical fibres. At the receiving end, a detector converts the light signals into electrical signals, from which the information can be recovered.