Refraction and Total Internal Reflection

Refraction and Total Internal Reflection

Refraction is the change in direction of a wave when it passes from one medium to another, caused by a change in wave speed. Total internal reflection (TIR) occurs when light travelling in a denser medium hits the boundary at an angle greater than the critical angle.

Snell's Law

When light passes from medium 1 to medium 2:

n₁ sin θ₁ = n₂ sin θ₂

where:

• n₁, n₂ = refractive indices of the two media
• θ₁ = angle of incidence (measured from the normal)
• θ₂ = angle of refraction

The refractive index of a medium: n = c/v

where c = speed of light in vacuum and v = speed in the medium.

For light going from air (n ≈ 1) into a medium of refractive index n:

n = sin θ₁ / sin θ₂

Worked Example

Light passes from air into glass (n = 1.52) at an angle of incidence of 40°. Find the angle of refraction.

sin θ₂ = sin 40° / 1.52 = 0.6428 / 1.52 = 0.4229

θ₂ = sin⁻¹(0.4229) = 25.0°

The light bends toward the normal because it slows down entering the denser medium.

Refraction and Wavelength

When light enters a denser medium:

• Speed decreases: v = c/n
• Wavelength decreases: λ_medium = λ_vacuum/n
• Frequency stays the same (frequency is set by the source)

This is why: v = fλ → if v decreases and f stays the same, λ must decrease.

Total Internal Reflection

When light travels from a denser medium (higher n) to a less dense medium (lower n), the refracted ray bends away from the normal. At a certain angle of incidence, the refracted ray travels along the boundary (θ₂ = 90°).

Critical angle, θ_c: The angle of incidence at which the refracted angle is exactly 90°.

From Snell's law: n₁ sin θ_c = n₂ sin 90° = n₂

sin θ_c = n₂/n₁

For glass-to-air: sin θ_c = 1/n_glass

Conditions for Total Internal Reflection

1. Light must be travelling from a denser to a less dense medium (n₁ > n₂)

2. The angle of incidence must be greater than the critical angle (θ > θ_c)

When θ > θ_c, all the light is reflected back into the denser medium — none is refracted.

Worked Example

Find the critical angle for a diamond-air boundary (n_diamond = 2.42).

sin θ_c = 1/2.42 = 0.4132

θ_c = sin⁻¹(0.4132) = 24.4°

This small critical angle is why diamonds sparkle — light entering the top undergoes multiple total internal reflections before emerging.

Optical Fibres

Optical fibres use total internal reflection to transmit light signals over long distances with minimal loss.

Structure:

• Core: Thin glass or plastic fibre (high refractive index, n₁)
• Cladding: Outer layer of glass with lower refractive index (n₂ < n₁)
• Protective sheath: Mechanical protection

Light enters the fibre and hits the core-cladding boundary at angles greater than the critical angle, undergoing TIR repeatedly along the length of the fibre.

Why Cladding is Necessary

1. Protects the core surface from scratches that would allow light to escape

2. Prevents cross-talk between adjacent fibres (light cannot leak from one core into another)

3. Ensures TIR occurs at a well-defined, controlled interface

4. Without cladding, the critical angle depends on whatever is touching the core (water, fingers, dirt)

Signal Degradation in Optical Fibres

Absorption: The glass absorbs some light energy, converting it to heat. Minimised by using ultra-pure glass; lowest absorption at wavelengths around 1.55 μm.

Pulse broadening (dispersion): Input pulses spread out over distance, potentially overlapping. Two main causes:

• Modal dispersion: Different rays take different paths (some travel straight, others bounce at steep angles, covering more distance). Reduced by using monomode fibres (very thin core, ~8 μm, allowing only one path).
• Material dispersion: Different wavelengths travel at slightly different speeds (n varies with λ). Reduced by using monochromatic sources (lasers rather than LEDs).

Repeaters/amplifiers are placed at intervals (~80 km) to regenerate the signal.

Applications of Optical Fibres

• Telecommunications: Broadband internet, telephone — high bandwidth, low loss, immune to electromagnetic interference
• Medical endoscopy: Viewing inside the body using a coherent fibre bundle (relative positions preserved)
• Sensors: Fibre optic sensors for temperature, pressure, strain

Applications of Total Internal Reflection

• Prisms in binoculars: 45-90-45° glass prisms reflect light by TIR (more efficient than mirrors, no silvering needed, since θ_c ≈ 42° for glass and the light hits at 45°)
• Bicycle reflectors and cat's eyes: Arrays of TIR prisms return light toward its source
• Diamond cutting: Maximises TIR for brilliance