Huygens' principle

Point a torch at a wall and the edge of the shadow is never perfectly sharp — light always bends a little into the dark region, as any wave does when it squeezes past an obstacle. Ray optics, with its straight-line rays and sharp shadows, cannot explain that bending, because ray optics is only ever an approximation of the deeper wave picture. Christiaan Huygens supplied the missing geometric machinery in 1678: treat every point on an advancing wavefront as the source of its own tiny spherical wavelet, and let the envelope of all those wavelets become the next wavefront. It sounds almost too simple to be powerful, yet this single construction rebuilds the laws of reflection and refraction from nothing but wave geometry — and, historically, it settled a two-century argument about whether light speeds up or slows down on entering glass.

The core insight

Every point on a wavefront is a source of secondary spherical wavelets travelling forward at the wave speed, and the new wavefront is the tangential envelope of all these wavelets — a construction that reproduces the laws of reflection and refraction purely from wave geometry.

The core idea

A wavefront is the locus of points in a medium vibrating in the same phase at a given instant. A point source in a uniform medium sends out spherical wavefronts (expanding concentric spheres); a source infinitely far away — like the Sun for practical purposes, or light after passing through a large converging lens — sends out wavefronts that are locally plane. In every case the direction of energy propagation, the "ray," is perpendicular to the wavefront at each point, so ray optics and wavefront geometry are two views of the same picture. Huygens' principle bundles two statements into one rule for building tomorrow's wavefront out of today's: Every point on an existing wavefront becomes a fresh source of secondary spherical wavelets, all spreading out in the forward direction with the same speed $v$ that the wave has in that medium. The new wavefront at a later instant is the surface tangential to all of these secondary wavelets — their common forward envelope. Only the forward tangent surface counts; a backward-facing envelope is not drawn, since it would predict a wave travelling back toward the source, which is not observed. Applied to a plane wavefront in a uniform medium, the construction simply reproduces another parallel plane wavefront further along — nothing changes direction. The construction only becomes physically informative at a boundary between two media, or at an obstacle, where the wave speed or the available secondary sources change. That is exactly where the two great proofs of ray optics come from.

Reflection. A plane wavefront $AB$ meets a plane mirror obliquely, striking $A$ first. While the wave still travelling in air covers the remaining distance from $B$ to a point $B'$ on the mirror in time $\tau$ , the secondary wavelet launched from $A$ has already expanded into a sphere of radius $v\tau$ in the same medium. The new reflected wavefront is the tangent plane drawn from $B'$ to that sphere. The two right triangles sharing hypotenuse $AB'$ then have equal legs $BB' = AA'' = v\tau$ (same medium, same speed on both sides), so the triangles are congruent and the angle the reflected wavefront makes with the mirror matches the angle the incident wavefront made — giving the familiar law $\angle i = \angle r$ , with incident ray, reflected ray and normal coplanar.

Refraction. The same picture is repeated across a boundary between two media of different wave speeds $v_1$ (incidence side) and $v_2$ (refraction side). In time $\tau$ , the part of the wavefront still in medium 1 advances a distance $v_1\tau$ to reach the surface, while the secondary wavelet from the point that arrived earlier has spread a distance $v_2\tau$ into medium 2. Reading the two right triangles that share the same hypotenuse across the interface gives:

\frac{\sin i}{\sin r}=\frac{v_1}{v_2}

where: $i$ is the angle of incidence, $r$ the angle of refraction, $v_1$ and $v_2$ are the wave speeds in the incidence and refraction media. Since refractive index is defined by $n=c/v$ , this ratio is equivalent to $n_1\sin i=n_2\sin r$ — Snell's law — but derived from wave geometry rather than assumed as an empirical rule.

Figure

A plane wavefront meeting an interface with a secondary wavelet expanding from the earlier point and a tangent line forming the new refracted wavefront at a different angle

Derivation ASnell's law from the Huygens construction

The Huygens construction for two media of different wave speed is the one grounded formula this section builds, and the goal is to show, with nothing beyond the geometry of two right triangles sharing a common hypotenuse across the interface, that the wave picture reproduces Snell's law and additionally predicts a definite, testable relationship between wave speed and refractive index that historically discriminated between the wave and corpuscular theories of light.

Step 1. Set up the two right triangles. Let the plane wavefront $AB$ reach the interface with $A$ touching first. While the ray through $B$ still needs time $\tau$ to reach the interface at $B'$ , covering distance $BB'=v_1\tau$ in medium 1, the point $A$ has already begun radiating its secondary wavelet into medium 2. Triangle $ABB'$ (with right angle at $B$ ) sits in medium 1; the new triangle $AA'B'$ (with right angle at $A'$ , where $A'$ is the tangent point of the wavelet) sits in medium 2. Both triangles share the same hypotenuse $AB'$ .

Step 2. Write the two sides in terms of the angles. In triangle $ABB'$ , angle $i$ is the angle between the incident wavefront's normal and the interface normal, and $BB'=AB'\sin i$ . In triangle $AA'B'$ , the secondary wavelet from $A$ has expanded to radius $A A'=v_2\tau$ in the time the wavefront needed to travel from $B$ to $B'$ , and by the geometry of the tangent construction $AA'=AB'\sin r$ , where $r$ is the angle the new wavefront makes with the interface on the refraction side.

Step 3. Also express $BB'=v_1\tau$ directly (distance travelled in medium 1 in time $\tau$ ), and $AA'=v_2\tau$ directly (distance the wavelet has spread in medium 2 in the same time $\tau$ ). Combine both expressions for $BB'$ : $AB'\sin i = v_1\tau$ , so $AB'=v_1\tau/\sin i$ . Combine both expressions for $AA'$ : $AB'\sin r = v_2\tau$ , so $AB'=v_2\tau/\sin r$ .

Step 4. Equate the two expressions for the common hypotenuse $AB'$ and cancel $\tau$ : $v_1\tau/\sin i = v_2\tau/\sin r$ , which rearranges directly to $\sin i/\sin r = v_1/v_2$ . Using $n=c/v$ this is $n_1\sin i=n_2\sin r$ , Snell's law, obtained purely from the wavefront-envelope construction. Because $\sin i/\sin r=v_1/v_2$ and denser media have larger $n$ (hence smaller $v$ ), the wave theory predicts that light travels slower, not faster, in a denser medium — the opposite of the older corpuscular theory's prediction, and a claim later confirmed experimentally by Foucault's speed-of-light measurements in water.

Worked example

A plane wavefront in air is incident on a plane mirror at 30 degrees to the normal, and separately, using the refraction relation $\sin i/\sin r=v_1/v_2$ , light passes from air into a medium of refractive index $n=1.6$ where the speed of light in air is $c=3\times10^{8}$ m/s.

Step 1. Reflection first. By the congruent-triangle result above, the law of reflection from Huygens' construction gives angle of incidence equal to angle of reflection, so the reflected ray also makes 30 degrees with the normal, i.e. 60 degrees with the mirror surface itself.

Step 2. The total angle turned through between the incident and reflected rays is $i+r=30\text{°}+30\text{°}=60\text{°}$ .

Step 3. Now the refraction part. Since $n=c/v_2$ with air as medium 1 ( $v_1=c$ ), rearranging gives the speed of light inside the medium:

v_2=\frac{c}{n}=\frac{3\times10^{8}}{1.6}=1.875\times10^{8}\ \text{m/s}

Step 4. Check the result is physically consistent with the derivation above: $v_2<v_1$ , exactly the "slower in a denser medium" prediction that the Huygens construction produces from wave geometry alone.

Answer: the reflected ray makes 30 degrees with the normal (60 degrees with the mirror), the total deviation between incident and reflected rays is 60 degrees, and the speed of light in the medium of index 1.6 is $1.875\times10^{8}$ m/s, slower than in air.

Common traps

Thinking Huygens' principle applies only to plane waves. It applies to any wavefront shape — spherical, plane, or otherwise; plane wavefronts are just the simplest case used for the standard reflection and refraction proofs.
Drawing secondary wavelets in every direction. Only the forward-facing tangent envelope is taken to build the new wavefront; the "why is there no backward wave" question is a known subtlety of the classical theory, not something to draw on an exam diagram.
Using Huygens' construction to derive prism or lens image formation. This syllabus explicitly restricts the construction to reflection and refraction at a plane interface; prism and lens results still come from ray-by-ray application of Snell's law, not from redoing the wavefront construction at curved surfaces.
Forgetting the historical significance of the speed prediction. The Huygens result $v_{\text{denser}}<v_{\text{light in air}}$ was a genuine testable prediction that distinguished the wave theory from the corpuscular theory (which predicted the opposite), later confirmed by Foucault — this is a common short-answer question.

Ask the tutor: want a step-by-step redraw of the reflection or refraction triangles, or a quick quiz on the wave-versus-corpuscular speed debate?

Why do we only take the forward envelope of the secondary wavelets and not the backward one too

Huygens' original construction only draws the tangent surface on the forward side because that is what matches observation — light does not travel back toward its own source after every wavefront update. Strictly, treating each point as an independent spherical source does not by itself forbid a backward wave; the deeper resolution requires the later, more complete wave theory (interference of secondary wavelets), which is beyond this syllabus. For exam purposes, simply state that only the forward envelope is used to construct the new wavefront.

Does Huygens construction let me derive how a prism bends light or how a lens focuses it

No, and this syllabus is explicit about that boundary. Huygens' principle here is applied only to plane or spherical wavefronts meeting a single plane interface, which is enough to reprove the laws of reflection and refraction. Prism deviation and lens image formation are ray-optics results built by applying Snell's law ray by ray through curved or angled surfaces; they are not re-derived from a fresh wavefront-envelope construction in this topic.

Why was the wave-theory prediction about the speed of light in glass historically important

Newton's corpuscular theory of light predicted that light should speed up when entering a denser medium like glass or water, since it imagined light particles being pulled in by the medium. Huygens' wave theory, through the construction $\sin i/\sin r=v_1/v_2$ , predicted the opposite: light should slow down in a denser medium. For nearly two centuries there was no direct way to measure the speed of light inside water, so the debate stayed unresolved. Foucault's mid-19th-century experiments finally measured light travelling slower in water than in air, confirming the wave theory's prediction and settling the historical dispute in Huygens' favour.

Huygens' principle

The core idea

Worked example

Common traps

Before this

Where this leads