Total Internal Reflection

When a wave passes from a denser medium into a less dense one, it bends away from the normal. As the angle of incidence increases, the angle of refraction grows even more rapidly. At a certain critical angle, the refracted ray would emerge along the boundary (angle of refraction = 90°). Beyond this, the wave simply cannot escape the denser medium: it is totally internally reflected back into the denser medium.

This phenomenon — total internal reflection (TIR) — is the operating principle of optical fibres, the technology that carries the majority of the world's internet traffic at the speed of light. It also underpins endoscopes, binoculars, fibre-optic illumination, prism reflectors and the sparkle of a well-cut diamond. The OCR A-Level Physics A specification (H556) requires you to derive and apply the critical-angle formula and to understand the physics of step-index optical fibres.

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The Critical Angle

Consider a ray of light travelling in medium 1 (refractive index n₁, larger) striking the boundary with medium 2 (refractive index n₂, smaller). Snell's law gives:

n₁ sin θ₁ = n₂ sin θ₂

As θ₁ increases, so does θ₂. At some specific angle — the critical angle θ_c — the refraction angle θ₂ reaches 90°. At this point, the refracted ray travels along the boundary itself and is said to be grazing. For angles of incidence greater than θ_c, no refracted ray can exist; all the light is reflected back into the denser medium.

Setting θ₂ = 90° and sin 90° = 1 in Snell's law:

n₁ sin θ_c = n₂ × 1

So:

sin θ_c = n₂ / n₁

This is the formula you must know. Some textbooks simplify it to sin θ_c = 1/n when the second medium is air (or vacuum), for which n₂ = 1. The more general form is the one above, and OCR expects you to handle both.

Conditions for Total Internal Reflection

TIR can occur if and only if:

1. The light is travelling from a medium of higher refractive index into one of lower refractive index (n₁ > n₂, otherwise sin θ_c > 1 which has no solution).
2. The angle of incidence is greater than the critical angle (θ₁ > θ_c).

Both conditions must be met. If the ray is going into a denser medium, no critical angle exists and TIR cannot happen.

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Worked Example 1 — Critical Angle for Glass-Air

Q. Calculate the critical angle for light travelling in glass (n = 1.52) meeting an interface with air (n = 1.00).

A.

sin θ_c = n₂ / n₁ = 1.00 / 1.52 = 0.658 θ_c = arcsin(0.658) = 41.1°

Any ray in the glass striking the glass-air surface at more than 41.1° from the normal will be totally internally reflected.

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Worked Example 2 — Critical Angle for Water-Air

Q. Calculate the critical angle for water (n = 1.33) meeting air.

A.

sin θ_c = 1.00 / 1.33 = 0.752 θ_c = 48.8°

An object underwater looking upwards sees the entire above-water world compressed into a cone of half-angle about 49° — Snell's window. Outside this cone, the surface acts as a mirror, reflecting the underwater scene.

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Worked Example 3 — Critical Angle Between Two Glasses

Q. Light travels from a core of refractive index 1.50 into a cladding of refractive index 1.47. Calculate the critical angle.

A.

sin θ_c = 1.47 / 1.50 = 0.980 θ_c = arcsin(0.980) = 78.5°

This is much larger than for a glass-air interface, because the two refractive indices are close. This is the characteristic geometry of a step-index optical fibre — only rays striking the core-cladding boundary at angles very close to grazing (>78.5° from the normal, or equivalently very small angles to the fibre axis) will be totally internally reflected and trapped in the core.

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Optical Fibres

An optical fibre is a long, thin strand of glass (or sometimes plastic) along which light is transmitted by repeated total internal reflection. Modern optical fibres are at the heart of telecommunications and medicine.

flowchart LR
    S[Light enters fibre] --> C1[Travels in core: n1 = 1.50]
    C1 --> B1[Strikes core-cladding boundary]
    B1 -- angle > theta_c --> TIR1[Total internal reflection]
    TIR1 --> C2[Back into core]
    C2 --> B2[Strikes other side]
    B2 -- angle > theta_c --> TIR2[TIR again]
    TIR2 --> E[Exits at far end]

Step-Index Fibre

A step-index fibre consists of:

• A core of high-refractive-index glass (typically n ≈ 1.50), where the light propagates.
• A cladding of slightly lower-refractive-index glass (typically n ≈ 1.47) surrounding the core.
• An outer protective coating (plastic jacket) for mechanical protection.

The refractive index "steps" abruptly at the core-cladding boundary (hence "step-index"). Light entering the core at a sufficiently shallow angle to the fibre axis will always strike the core-cladding boundary at an angle greater than the critical angle, and so will be totally internally reflected. The light is trapped in the core and travels along the fibre with almost no loss.

Why Cladding?

You might wonder why you cannot just use an uncoated glass core with air outside. There are two essential reasons for the cladding.

1. Protection of the total-reflection surface. If the glass surface is touched, scratched or contaminated by dust, grease or moisture, TIR breaks down at those points and light leaks out. The cladding protects the core from surface contamination by keeping the TIR surface inside the fibre.

2. Prevention of cross-talk between fibres. If two uncoated fibres touched, light could cross from one to the other (the fibres would act as a single enlarged core at the contact point). Cladding keeps each fibre's light safely contained.