Huygens' Principle

Huygens' principle is the geometric heart of wave optics. It explains how an arbitrary wavefront evolves in time and underlies the derivations of reflection, refraction, interference, and diffraction.

Concept

Wavefront. A surface of constant phase. Examples:

A point source produces spherical wavefronts.
A distant source produces plane wavefronts (effectively flat over a small region).
A line source produces cylindrical wavefronts.

Huygens' principle. Every point on a wavefront can be regarded as a secondary source emitting spherical wavelets. The new wavefront at a later time is the forward envelope of all these secondary wavelets.

The (Huygens-)Fresnel-Kirchhoff refinement adds that the amplitude of a secondary wavelet is greatest in the forward direction and zero backward — accounting for why light does not "go backwards" by this construction alone.

Construction

To find the wavefront at time $t + \Delta t$ given the wavefront at time $t$ :

Pick many points on the existing wavefront.
From each, draw a sphere of radius $c\Delta t / n$ (the distance the wave travels in that medium in $\Delta t$ ).
Draw the surface tangent to all these spheres on the forward side. This is the new wavefront.

For a plane wave in a uniform medium, the construction reproduces a parallel plane shifted by $c\Delta t$ .

For a spherical wave from a point source, it gives a larger sphere centred on the source.

Why It Works

The wave equation is linear: a superposition of point-source solutions is still a solution. Each point on a wavefront serves as a point source with the same phase. The envelope captures the dominant constructive interference of all wavelets in the forward direction; destructive interference cancels them elsewhere (essentially the principle of stationary phase).

Worked Example

A plane wavefront in air strikes the flat surface of glass head-on. Using Huygens:

Each point on the wavefront enters the glass and emits secondary wavelets with smaller radius (speed in glass $v_g = c/n < c$ ).
The envelope inside the glass is a plane parallel to the original, but shifted by less distance in the same time.
Outside the glass, the original wavelet speed $c$ is preserved.

This construction is used to derive Snell's law for an oblique incidence (see next topic).

Common Confusions

Huygens' construction is not just "draw circles" — the envelope is the new wavefront, not the wavelets themselves.
The construction predicts a backward envelope too, which would imply the wave goes backwards. Fresnel resolved this by including an angular factor (obliquity factor) that suppresses backward radiation.
Wavelets propagate at the wave's speed in the current medium, not in vacuum.

Key Takeaways

A wavefront is a surface of constant phase.
Huygens: each point of a wavefront acts as a source of secondary wavelets; the envelope is the new wavefront.
Reproduces plane and spherical wave propagation.
Forms the geometric basis for reflection, refraction, and diffraction.