Why Light Slows Down and Bends When It Enters Glass

Ray optics explains mirrors and lenses with straight lines, but it cannot say why light bends on entering glass or why it spreads round edges. Treating light as a wave, built up from tiny secondary wavelets, answers both.

This part covers wavefronts and Huygens principle, and uses them to prove the laws of reflection and refraction.

A wavefront is a surface on which every point oscillates in the same phase, and Huygens principle says every point on a wavefront acts as a source of secondary wavelets whose forward common tangent, a short time later, is the new wavefront.

Shapes of wavefronts:

- Spherical — from a point source
- Cylindrical — from a line source such as a slit
- Plane — far from any source, as with sunlight reaching the Earth

Rays are lines perpendicular to the wavefronts, showing the direction in which energy travels.

Huygens construction:

- Treat each point of the wavefront as a new source
- In time $t$, each wavelet spreads to a radius $vt$
- Draw the forward envelope of these wavelets — that is the new wavefront; the backward wave is ignored

Worked example. In $10^{-10}$ s, wavelets travel $3\times 10^8\times 10^{-10}=0.03$ m $=3$ cm in air, but in glass of $n=1.5$, where $v=\frac{c}{n}=2\times 10^8$ m s${}^{-1}$, they travel only $2$ cm.

An everyday example. Ripples from a stone dropped in a still pond spread as circular wavefronts, and far from the stone a short stretch of ripple looks almost straight — a plane wavefront.

The substance. Wavefronts and rays are always perpendicular — so the bending of a wavefront is the bending of a ray.

When a plane wavefront strikes a mirror, the wavelet from the point that arrives first grows to the same radius as the distance the rest of the wavefront still has to travel, and the congruent triangles this creates prove that the angle of reflection equals the angle of incidence.

The proof:

- A plane wavefront AB meets the mirror at A, at angle of incidence $i$
- In time $t$, point B travels to C on the mirror: $BC=vt$
- Meanwhile the wavelet from A grows to radius $AE=vt$
- The tangent CE from C to this wavelet is the reflected wavefront
- Triangles ABC and AEC have right angles at B and E, a common hypotenuse AC, and $BC=AE$, so they are congruent

$$\sin i=\frac{BC}{AC}=\frac{vt}{AC}=\frac{AE}{AC}=\sin r\Rightarrow i=r$$

Worked example. A wavefront meets a mirror at $i=30^{\circ }$ with $AC=10$ cm:

$$BC=AC\sin i=10\times 0.5=5\text{ cm},t=\frac{0.05}{3\times 10^8}\approx 1.7\times 10^{-10}\text{ s}$$

The wavelet from A reaches $AE=5$ cm, so $\sin r=\frac{5}{10}=0.5$ and $r=30^{\circ }$.

An everyday example. A still lake at dawn mirrors the trees on its bank because each wavefront of light is reflected in exactly this way.

The substance. The speed is the same before and after reflection, so the wavelength does not change either.

**When a plane wavefront enters a slower medium, its wavelets there spread a shorter distance in the same time, so the wavefront turns; comparing the two distances gives $\frac{\sin i}{\sin r}=\frac{v_1}{v_2}=\frac{n_2}{n_1}$, which is Snell's law.

The proof:**

- A plane wavefront AB meets the boundary at A, at angle $i$
- In time $t$, point B travels to C in medium 1: $BC=v_{1}t$
- The wavelet from A spreads in medium 2 to radius $AE=v_{2}t$
- The tangent CE is the refracted wavefront

$$\sin i=\frac{v_{1}t}{AC},\sin r=\frac{v_{2}t}{AC}\Rightarrow \frac{\sin i}{\sin r}=\frac{v_1}{v_2}=\frac{n_2}{n_1}$$

since $n=\frac{c}{v}$. The frequency stays the same, so the wavelength changes in proportion to speed: ${\lambda }^{\prime }=\frac{\lambda }{n}$ when light enters a medium of index $n$ from air.

Worked example. A wavefront enters glass ($n=1.5$) from air at $i=45^{\circ }$, with $AC=10$ cm:

$$BC=10\times 0.707\approx 7.07\text{ cm},AE=\frac{7.07}{1.5}\approx 4.71\text{ cm},\sin r=0.471\Rightarrow r\approx 28^{\circ }$$

Light of wavelength $600$ nm in air has wavelength $\frac{600}{1.5}=400$ nm in the glass.

An everyday example. A row of people walking arm in arm from a road onto beach sand at an angle swings round, because those reaching the sand first slow down first — just as a wavefront turns.

The substance. The proof works only if light travels more slowly in the denser medium, and measurements confirm that it does.