Physical Optics and Huygens' Principle

Hold a drinking straw at an angle in a glass of water and it looks snapped in two at the surface. School optics explains the bend with a rule you memorise — n_1\sin\theta_1 = n_2\sin\theta_2 — and leaves it there. But why should light bend at all? Why exactly that ratio of sines, and not some other law? The ray picture can only assert the rule; it cannot explain it. To explain it we have to stop drawing light as an arrow and start treating it as what it is: a wave.

The single idea on this page is Huygens' principle — a beautifully simple recipe for how a wavefront moves forward that a Dutch astronomer wrote down in 1678, long before anyone knew what light was made of. From that one recipe, the whole of ray optics falls out: the law of reflection and the law of refraction are not separate rules to memorise but consequences of how wavefronts advance. This is the doorway into physical optics — optics that takes the wave seriously — and everything later (interference, diffraction, the Fourier optics of a lens) is built on it.

Wavefronts and rays

Drop a pebble in a still pond and circular ripples spread outward. A wavefront is a surface of constant phase — the crest of one ripple, say — so the ripples are a family of wavefronts, each a circle a little larger than the last. Far from the pebble the circles are so big that a small patch of one looks straight: a plane wave. A ray is just the direction the wavefront is marching, always at right angles to it. Rays and wavefronts are two views of the same thing: the ray tells you where the light goes, the wavefront tells you when it gets there.

Ray optics tracks the arrows and forgets the surfaces. Physical optics keeps the surfaces — and that is what lets it explain, rather than merely state, how light behaves at a boundary.

Huygens' principle: every point is a new source

Here is the recipe. Take a wavefront at some instant. Now imagine that every point on it is a tiny source that sends out its own little spherical wave — a secondary wavelet — travelling forward at the wave speed of the medium. Let a short time t pass; each wavelet has grown into a sphere of radius v\,t. The new wavefront is simply the surface that just touches all of these wavelets at once — their common tangent, or envelope.

For a plane wave in a uniform medium this just reproduces the obvious: a flat sheet of wavelets, all the same radius, has a flat envelope a distance v\,t further on. The wave marches straight ahead. The magic appears when the wave crosses into a medium where v is different, or meets an obstacle — then the wavelets have different radii, the envelope tilts, and the wave bends. Let us watch exactly that.

Refraction, derived — not memorised

A plane wavefront in air drifts down onto a slab of glass at an angle. Call the two ends of the wavefront A and B. Because the wavefront is tilted, one end — say A — touches the glass first while the other end is still up in the air. During the time t it takes the far corner to cover the last stretch BB'=v_1 t and reach the surface, the corner already inside the glass has been busy launching its wavelet. But glass is slower (v_2 < v_1), so in that same time its wavelet has only grown to the smaller radius AD = v_2 t. The new wavefront is the line from B' tangent to this stunted wavelet — and because that wavelet is smaller, the wavefront is tilted toward the normal. Step through the construction:

Now read the geometry. Both right triangles are built on the same hypotenuse AB' along the surface. In the upper triangle the side opposite \theta_1 is BB'=v_1 t; in the lower one the side opposite \theta_2 is AD=v_2 t. So

AB' = \frac{v_1 t}{\sin\theta_1} = \frac{v_2 t}{\sin\theta_2} \quad\Longrightarrow\quad \frac{\sin\theta_1}{\sin\theta_2} = \frac{v_1}{v_2}.

The refractive index is defined as n = c/v — how many times slower light goes in the medium than in vacuum — so v_1/v_2 = n_2/n_1, and the ratio of sines becomes the law you were once asked to take on faith:

The identical construction with the wavelet on the same side as the incoming wave gives the law of reflection: both corners are in the same medium, so both wavelets grow at the same speed, the two triangles are congruent, and \theta_i = \theta_r. Two of school optics' hard rules, both squeezed out of one idea about advancing wavelets.

A worked example

Example — a ray entering glass. Light in air (n_1 = 1.00) strikes a glass block (n_2 = 1.50) at an angle of incidence \theta_1 = 30^\circ. Find the refraction angle. Snell's law gives

\sin\theta_2 = \frac{n_1}{n_2}\sin\theta_1 = \frac{1.00}{1.50}\sin 30^\circ = \frac{0.500}{1.50} = 0.333,

so \theta_2 = \arcsin(0.333) \approx 19.5^\circ. The ray has bent toward the normal, as promised for the slower medium — from 30^\circ down to about 19.5^\circ.

Run it backwards. Inside the glass a ray hits the far face at \theta = 45^\circ trying to get back out into air. Now \sin\theta_{\text{air}} = 1.50\sin 45^\circ = 1.06 — which is impossible, since a sine can never exceed 1. There is no refracted ray: the light is thrown entirely back into the glass. Huygens' bookkeeping has just predicted total internal reflection, the effect that pipes light down an optical fibre.

Huygens' wave theory made a bold, testable claim: because the ray bends toward the normal in glass, the construction only works if light travels slower in the denser medium (v_2 < v_1). Newton's rival corpuscular theory — light as a stream of particles pulled into the glass — required the exact opposite, that light speeds up inside. For over 150 years no one could measure light fast enough to tell them apart, and Newton's enormous prestige kept the particle picture on top. Then in 1850 Léon Foucault measured the speed of light directly in water and found it slower than in air. Huygens, dead for 155 years, was right; and the wave theory reigned — until the photon complicated the story all over again a half-century later.

Two traps snare almost everyone. First: angles are measured from the normal, not the surface. Snell's law uses the angle between the ray and the perpendicular to the boundary — plug in the glancing angle to the surface and every answer comes out wrong. Second: Huygens' wavelets go forward only. A literal reading of "every point radiates a full sphere" would have half of each wavelet travelling backward, predicting a reflected wave even in open space where there is none. Huygens simply asserted the backward halves cancel; it took Fresnel and Kirchhoff, a century and a half later, to show why — the wavelets carry a phase and a direction-dependent strength (an "obliquity factor") that suppresses the backward wave through interference. The simple recipe works, but the honest justification needs the full wave equation.