Total internal reflection
The critical angle
Shine a laser pointer upwards through a tank of water, towards the surface. Most of the light escapes into the air above, but a faint beam is also reflected back down.
Going from water, the denser medium, into air, the rarer medium, the ray bends away from the normal. So the angle of refraction r is always bigger than the angle of incidence i.
Now tilt the laser so that i increases. r increases faster. The refracted beam swings down towards the surface and gets fainter, while the reflected beam gets brighter.
At one particular angle of incidence, the refracted ray just grazes along the surface: r is 90°. This angle of incidence is the critical angle, i c.
Tilt a little more, beyond the critical angle. There is no refracted ray at all. Every bit of the light is reflected back into the water. This is total internal reflection.
Two conditions are needed. Light must travel from a denser to a rarer medium, and the angle of incidence must be greater than the critical angle.
To find i c, put r = 90° in Snell’s law. n₁ sin i c equals n₂ sin 90°, which is n₂. So sin i c = n₂/n₁. For a medium with air outside, sin i c = 1/n.
For glass of refractive index 1.5, sin i c is 1 over 1.5, or 0.667, so the critical angle is about 41.8°. For water, 1.33, it is about 48.8°.
The denser the material, the smaller the critical angle. Diamond, with n = 2.42, has a critical angle of only 24.4°.
Diamonds, prisms and optical fibres
Total internal reflection is used all around us. Start with a diamond. Its critical angle is just 24.4°, so light that enters is very likely to be trapped.
A diamond cutter shapes the faces so that light bounces inside again and again by total internal reflection before it escapes from the top. That is why a well-cut diamond sparkles.
Next, a right-angled glass prism with angles of 45°, 90° and 45°. Light enters one short face straight on, and meets the long face at 45°.
45° is more than the critical angle of glass, 41.8°. So the light is totally reflected and leaves through the other short face, turned through 90°.
Turn the prism so the light enters the long face. Now it reflects twice and comes back, turned through 180°. Such prisms bend or invert light in periscopes and binoculars, better than silvered mirrors, because nothing is lost.
Finally, the optical fibre. It is a thin strand of glass, the core, covered by a layer of glass of lower refractive index, the cladding.
Light enters one end and strikes the core–cladding boundary at an angle greater than the critical angle. It is totally reflected, again and again, zig-zagging along the fibre.
Even when the fibre bends gently, the light stays trapped and comes out of the far end with very little loss.
Bundles of fibres carry pictures from inside the body in an endoscope. They also carry telephone calls and internet data as pulses of light, across cities and under oceans.
Try it: find the critical angle
- Glass → air. Raise i slowly: the refracted ray swings towards the surface and fades.
- The faint dashed ray marks the critical angle c. Beyond it, the refracted ray disappears and 100% is reflected.
- Try Diamond → air. The critical angle drops to about 24°.
- Press Swap media (air → glass). Can you get total internal reflection now?
Try it: light in an optical fibre
- Drag the amber dot to change the angle at which light enters the fibre.
- Compare the angle at the wall φ with the critical angle c. When φ > c, the light is trapped.
- Increase the entry angle until light starts leaking through the cladding.
- Lower the cladding index n₂. The critical angle falls, so the fibre accepts light over a wider cone.
Critical angle of water
What is the critical angle for light going from water (n = 1.33) into air?
- 41.8°
- 48.8°
- 53.1°
- 36.9°
Answer
B. sin ic = n₂/n₁ = 1/1.33 = 0.752, so ic ≈ 48.8°. (41.8° is the value for glass of n = 1.5.)
Glass under water
Light inside glass (n = 1.5) strikes a glass–water boundary (water n = 1.33) at an angle of incidence of 60°. What happens?
- Total internal reflection, because 60° is more than 41.8°
- It refracts into the water, because the critical angle here is about 62.5°
- Total internal reflection always happens when light leaves glass
- It goes straight through without bending
Answer
B. The critical angle depends on both media: sin ic = 1.33/1.5 = 0.887, so ic ≈ 62.5°. Since 60° < 62.5°, the light refracts into the water (sin r = 1.5 × sin 60°/1.33 = 0.977, r ≈ 77.6°). 41.8° is the critical angle for glass to air.
Summary
- Critical angle:
sin i_c = n₂/n₁ - Into air:
sin i_c = 1/n
- Total internal reflection needs light going from a denser to a rarer medium, at an angle of incidence greater than the critical angle.
- At the critical angle the refracted ray grazes the surface (r = 90°). Beyond it, all the light is reflected.
- Critical angles: water ≈ 48.8°, glass (1.5) ≈ 41.8°, diamond ≈ 24.4°.
- Uses: sparkle of diamonds, totally reflecting prisms (90° and 180° turns in periscopes and binoculars), and optical fibres (endoscopes, telecommunication).