Physics · Ch 10 — Wave Optics

Refraction of a Plane Wave by Huygens' Construction

10.4

Refraction of a Plane Wave by Huygens' Construction

The same construction, applied at a refracting interface separating two media of speeds $v_1$ (medium 1, where the wave is incident) and $v_2$ (medium 2, where it is transmitted), yields Snell's law. Let the plane wavefront $AB$ be incident on the plane interface $MN$ at angle $i$ , with $A$ already on the interface. In the time $\tau$ taken for point $B$ to reach the interface at $C$ , so that $BC=v_1\tau$ , the point $A$ has been sending out a secondary wavelet into the second medium, which by Huygens' principle spreads with speed $v_2$ and has therefore grown to radius $AE=v_2\tau$ by the time $C$ is reached. The refracted wavefront is the tangent $CE$ drawn from $C$ to this secondary wavelet, making an angle of refraction $r$ with the interface normal at $A$ .

In the right triangle $ABC$ , $\sin i=BC/AC=v_1\tau/AC$ , and in the right triangle $AEC$ , $\sin r=AE/AC=v_2\tau/AC$ . Dividing the first relation by the second eliminates the common factor $\tau/AC$ and gives

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

Figure 1Huygens' construction proving Snell's law of refraction at a plane interface

What this figure shows. A ray/wavefront diagram showing a plane interface MN drawn as a horizontal line separating an upper, less dense medium 1 (speed v1) from a lower, denser medium 2 (speed v2 < v1). An incident plane wavefront AB in medium 1 is shown touching the interface at point A while its other end B is still above the interface, with arrows showing its direction of travel down towards the interface, and the angle of incidence i marked between the incident wavefront's direction and the dashed normal drawn perpendicular to MN at A. A second point C is marked on the interface to the right of A, with the straight segment BC = v1tau drawn in medium 1. Centred on A, a semicircular arc of radius AE = v2tau is drawn below the interface, in medium 2, representing the secondary wavelet spreading from A into the denser medium; point E lies on this arc such that CE is tangent to the arc. The straight line CE is drawn as the refracted wavefront in medium 2, bent visibly closer to the normal than the incident wavefront was, with arrows showing its direction of travel, and the angle of refraction r (smaller than i) marked between CE and the normal continued below the interface. The two right triang …

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