Total internal reflection — critical angle and applications in optical fibre

When light travels from a denser medium to a rarer medium (e.g., glass to air), and the angle of incidence exceeds a certain value called the critical angle, all light is reflected back into the denser medium. No refraction occurs — this is total internal reflection.

Two conditions for TIR: (1) Light must go from denser to rarer medium. (2) Angle of incidence must exceed the critical angle $\theta_c$.

At the critical angle, the refracted ray grazes along the interface (angle of refraction = 90°). Applying Snell's law:

$n_1 \sin\theta_c = n_2 \sin 90°$

$n_1 \sin\theta_c = n_2$

$\sin\theta_c = \frac{n_2}{n_1}$

For glass ($n_1 = 1.5$) to air ($n_2 = 1$):

$\theta_c = \sin^{-1}\left(\frac{1}{1.5}\right) = \sin^{-1}(0.667) \approx \mathbf{42°}$

An optical fibre consists of a core (higher refractive index) surrounded by cladding (lower refractive index). Light enters the core and hits the core-cladding boundary at an angle greater than the critical angle.

At each reflection point, TIR occurs — no light escapes through the cladding. The light bounces along the fibre, following even curved paths. This allows data transmission over kilometres with minimal signal loss.

• Core: $n_1 \approx 1.62$ (higher)
• Cladding: $n_2 \approx 1.52$ (lower)
• Acceptance angle: The maximum angle at which light can enter the fibre and still undergo TIR inside

The small difference in refractive indices is carefully engineered — too large and the fibre becomes multimodal (signal dispersion); too small and TIR becomes unreliable.