Refraction at Plane Surfaces

When light travels from one transparent medium into another — say, from air into water — it changes speed. This change in speed causes the light ray to bend at the boundary between the two media. This bending is called refraction. You have seen it countless times: a straw in a glass of water appears bent at the surface, or a swimming pool looks shallower than it really is. Refraction is not an optical illusion; it is a real physical phenomenon arising from the wave nature of light and the different speeds at which light propagates in different materials. The key quantity that governs refraction is the refractive index, n, of a medium. It is defined as the ratio of the speed of light in a vacuum, c ≈ 3.00 × 10⁸ m s⁻¹, to the speed of light in the medium, v. A higher refractive index means light travels more slowly in that medium. Air has n ≈ 1.00, water n ≈ 1.33, and glass typically n ≈ 1.5. When light enters a denser medium (higher n) it slows down and bends towards the normal — the imaginary line perpendicular to the surface at the point of incidence. The precise mathematical relationship governing refraction at a plane (flat) surface is Snell's Law, independently discovered by Willebrord Snell and later derived from wave theory by Christiaan Huygens. It connects the angles of incidence and refraction to the refractive indices of the two media and is the foundation of almost all optical instrument design, from camera lenses to fibre-optic cables.

Key takeaways

The Normal and Angle Convention
Why Light Bends: A Wave Picture
Snell's Law in Practice
Apparent Depth
Total Internal Reflection

The Normal and Angle Convention

All angles in refraction are measured from the normal — a line drawn perpendicular to the interface at the exact point where the ray meets the surface. The angle of incidence θ₁ is the angle between the incoming ray and the normal on the incident side. The angle of refraction θ₂ is the angle between the refracted ray and the normal on the transmitted side. This convention keeps mathematics consistent regardless of the orientation of the surface.

Why Light Bends: A Wave Picture

Think of a marching band walking in formation on a smooth floor and then crossing onto grass at an angle. The side of the band that hits the grass first slows down while the other side is still on smooth floor, causing the whole formation to pivot — to change direction. Light waves behave identically: the part of the wavefront entering the slower medium first slows down, and the wavefront pivots. The ray, always perpendicular to the wavefront, bends accordingly. Entering a slower medium (higher n) bends the ray towards the normal; entering a faster medium (lower n) bends it away.

Snell's Law in Practice

Snell's Law states n₁ sin θ₁ = n₂ sin θ₂. To use it: identify the refractive indices of both media, measure or note the angle of incidence, then solve algebraically for the unknown angle or refractive index. If n₂ > n₁ then sin θ₂ < sin θ₁, so θ₂ < θ₁ — the ray bends towards the normal. If n₂ < n₁ then θ₂ > θ₁ — the ray bends away. When light travels along the normal (θ₁ = 0°) it passes straight through without bending, regardless of the refractive indices.

Apparent Depth

A practical consequence of refraction at plane surfaces is apparent depth. An object at real depth d beneath a flat water surface appears to be at a shallower apparent depth d' = d / n when viewed from directly above (using the paraxial approximation for small angles). For water with n = 1.33, an object 1.33 m deep appears to be only 1.00 m deep. This explains why rivers and pools always look shallower than they truly are — a safety consideration that swimmers and divers must keep in mind.

Total Internal Reflection

When light travels from a denser medium into a less dense one (n₁ > n₂), θ₂ > θ₁. As θ₁ increases, a point is reached where θ₂ = 90° — the refracted ray travels along the surface. The angle of incidence at this point is the critical angle θ_c, given by sin θ_c = n₂/n₁. For angles of incidence greater than θ_c, no refracted ray exists: all the light is reflected back into the denser medium. This phenomenon, total internal reflection, is the principle behind optical fibres and the brilliance of cut diamonds.

The Key Equation

n_1 \sin\theta_1 = n_2 \sin\theta_2

n_1 — Refractive index of medium 1 (dimensionless): The ratio c/v for the medium containing the incident ray; equals 1.00 for air.
\theta_1 — Angle of incidence (degrees or radians): Angle between the incident ray and the normal to the surface at the point of incidence.
n_2 — Refractive index of medium 2 (dimensionless): The ratio c/v for the medium containing the refracted ray.
\theta_2 — Angle of refraction (degrees or radians): Angle between the refracted ray and the normal on the transmitted side of the surface.

Worked Example

Problem. A ray of light in air (n₁ = 1.00) strikes the flat surface of a glass block (n₂ = 1.50) at an angle of incidence of 40°. Find the angle of refraction inside the glass.

\theta_2 = \sin^{-1}\!\left(\frac{1.00}{1.50}\sin 40^\circ\right) \approx 25.4^\circ

Answer. The angle of refraction is approximately 25.4°.

Figure

Diagram

Origins · 1 minWillebrord Snellius (Snell) and how this was discovered

Willebrord Snellius (Snell)

1580–1626 · Dutch

Willebrord Snell, a Dutch mathematician and astronomer at the University of Leiden, discovered the law of refraction empirically around 1621 through careful geometric measurements of light passing through media. Although he did not publish the result himself, it was described after his death and later popularised by René Descartes in 1637. The law, now universally called Snell's Law, provides the exact mathematical relationship between angles of incidence and refraction and the refractive indices of the media involved. Snell was also a pioneer of trigonometric surveying and introduced the method of triangulation to accurately measure large distances on Earth — an entirely different practical application of his geometric genius.

Reflection at Plane Surfaces — Reflection and refraction both occur at an interface; understanding the law of reflection provides the contrast needed to fully appreciate why refracted rays behave differently from reflected ones.
Total Internal Reflection — Total internal reflection is a direct consequence of Snell's Law when light travels from a denser to a less dense medium and the angle of incidence exceeds the critical angle.
Refraction at Curved Surfaces — Curved surfaces (lenses) apply Snell's Law locally at each point; understanding plane surface refraction is the essential prerequisite for analysing lenses and optical instruments.
Wave Nature of Light — Refraction arises because light is a wave; Huygens' wave model explains why the wavefront pivots when it crosses a boundary, providing the physical basis for Snell's Law.