Q.(a) Using Huygens' construction, draw a figure showing the propagation of plane wave refracting at a plane surface separating two media. Hence, verify Snell's law of refraction. (1+2)

Huygens' principle constructs the refracted wavefront from secondary wavelets, whose geometry directly reproduces Snell's law; passing unpolarised light through 3 polarisers each rotated 30° from the previous one transmits 9/32 of the original intensity.

1. Huygens' construction and Snell's law (described): Consider a plane wavefront AB incident obliquely on a plane interface XY separating medium 1 (speed $v_1$) from medium 2 (speed $v_2$), making angle of incidence $i$ with the normal. According to Huygens' principle, every point on the wavefront acts as a source of secondary spherical wavelets that spread out with the speed appropriate to the medium they are in. At the instant point A on the wavefront touches the interface, point B (also on the incident wavefront) still has to travel a distance $BC$ (at speed $v_1$) to reach the interface, taking time $\tau = BC/v_1$. During this same time $\tau$, the secondary wavelet from A (already at the interface, now inside medium 2) spreads out to a sphere of radius $AD = v_2\tau$. The new refracted wavefront is the common tangent CD drawn from C to this expanding wavelet. From the right triangles $ABC$ (angle $i$ at A) and $ACD$ (angle $r$ at C, the angle of refraction), with $AC$ as the common hypotenuse: $\sin i = \dfrac{BC}{AC} = \dfrac{v_1\tau}{AC}, \qquad \sin r = \dfrac{AD}{AC} = \dfrac{v_2\tau}{AC}$ Dividing: $\dfrac{\sin i}{\sin r} = \dfrac{v_1}{v_2} = n_{21}$ This is Snell's law of refraction, where $n_{21}$ is the refractive index of medium 2 relative to medium 1 — derived purely from the geometry of Huygens' wavelets.
2. Fraction of light transmitted through three polarisers: …