Reflection & Refraction

When refraction stops working

• As the angle of incidence inside a dense medium is gradually increased, the angle of refraction in the surrounding less-dense medium also increases, but it grows faster than i, because the ray is bending away from the normal as it leaves
• Eventually, at one particular incident angle, the refracted ray bends right round to lie flat along the boundary (refraction angle = 90°). This incident angle is called the critical angle, c
• Any incident angle larger than the critical angle can no longer produce a refracted ray at all; all the light is instead reflected back inside the dense medium

Definition

• Total internal reflection (TIR) is the complete reflection of a wave back into its original medium when:
  • the ray is travelling from a denser medium into a less dense one (so refraction would bend it away from the normal), and
  • the angle of incidence is greater than the critical angle
• Both conditions are needed; if the light is travelling from less dense to denser, no critical angle exists

Critical angle and refractive index

sin c = 1 / n

• Rearranges to:
  • c = sin⁻¹(1 / n)
  • n = 1 / sin c
• A consequence: the larger the refractive index, the smaller the critical angle, and the more easily light gets trapped by TIR. This is exactly why cut diamonds (n = 2.42) sparkle so brightly; most rays striking the diamond's back surfaces have angles bigger than the critical angle and bounce repeatedly inside the gem before emerging

Example — calculate the critical angle for the boundary between water (n = 1.33) and air.

• sin c = 1 / n = 1 / 1.33 = 0.7519
• c = sin⁻¹(0.7519) = 48.8°
• Underwater rays that hit the surface at angles steeper than 48.8° to the normal cannot escape; they reflect back down, which is why a swimmer looking up sees a mirror beyond a certain angle

Useful applications of TIR

• Optical fibres:
  • A long, thin core of high-refractive-index glass (or plastic) is surrounded by a slightly lower-index outer cladding
  • Light injected at one end of the fibre keeps striking the core–cladding boundary at angles bigger than the critical angle, so it total-internal-reflects along the entire length, even round bends
  • Applications:
    • telecommunications: fibre-optic cables carry the world's internet traffic at near-light speed, using infrared pulses
    • medical endoscopes: a flexible bundle of fibres carries an image of the inside of a patient's body out through a small incision
    • decorative lighting: fibre lamps glow at the tip from a light source at the base
• Right-angled prisms:
  • A right-angled prism cut from glass has a critical angle of about 42°; a ray striking its long face at 45° therefore undergoes TIR
  • Applications:
    • periscopes: a vertical tube with a right-angled prism at the top and another at the bottom turns light through 90° twice, letting a viewer below see what is above the obstruction
    • binoculars and telescopes: internal prisms fold the light path so the instrument can be made short and still have a long total path
    • bicycle and road safety reflectors: built from tiny corner-cube prisms that bounce headlight light straight back to the source