Nature and Propagation of Light

a) Spherical to plane wavefront

A point source produces spherical wavefronts. If the point source is placed at the focus of a convex lens (or of a concave mirror), the rays become parallel after refraction, so the wavefront becomes plane.

            spherical         convex        plane
            wavefronts         lens       wavefronts
                (  (           /\          |  |  |
      S •      (  (  (        |  |  ---->  |  |  |
     (at F)     (  (           \/          |  |  |
          |<----- f ----->|

The parts of the spherical wavefront near the axis travel through the thicker middle of the lens and are slowed more, while the outer parts travel through thinner glass. The curved wavefront is straightened into a plane one.

b) Law of reflection from Huygens' principle

     incident rays          reflected rays
          \     \             /     /
           \     B           D     /
            \   / \         / \   /
             \ /   \       /   \ /
   X __________A_____________C__________ Y
              (AB = incident wavefront,  CD = reflected wavefront,
               BC = AD = vt)

Let a plane wavefront $AB$ strike the reflecting surface $XY$ at angle $i$. When end $A$ touches the surface, end $B$ is still a distance $BC$ away from it.

1. In the time $t$ that the disturbance at $B$ travels to $C$, $BC = vt$.
2. In the same time, the secondary wavelet from $A$ spreads as a hemisphere of radius $AD = vt$ (same medium, same speed).
3. Draw the tangent $CD$ from $C$ to this wavelet. By Huygens' principle, $CD$ is the reflected wavefront.
4. In right-angled triangles $ABC$ and $ADC$ (right angles at $B$ and $D$):
   • $AC$ is common,
   • $BC = AD = vt$. So $\triangle ABC \cong \triangle CDA$ (RHS).
5. Hence $\angle BAC = \angle DCA$. But $\angle BAC = i$ (angle between incident wavefront and surface) and $\angle DCA = r$ (angle between reflected wavefront and surface).

$$\boxed{\angle i = \angle r}$$

6. The incident wavefront, reflected wavefront and the normal all lie in the plane of the paper.

These are the laws of reflection.

A plane wavefront (parallel beam) can be converted into a spherical wavefront by passing it through a convex lens (or reflecting it from a concave mirror).

   plane         convex       converging spherical
 wavefronts       lens          wavefronts
  |  |  |          /\            )  )
  |  |  |  ---->  |  |  ---->   ) )  •F
  |  |  |          \/            )  )

Justification: The central part of the plane wavefront passes through the thick middle of the lens, where light travels slowly, so it is delayed most. The edges pass through thin glass and are delayed less. The wavefront comes out curved and converges to the focus $F$, i.e. it becomes a converging spherical wavefront centred at $F$. A concave lens does the opposite and gives a diverging spherical wavefront that appears to come from its focus.

(C) polarization

Huygens took light waves as longitudinal waves. Polarization is possible only for transverse waves, so Huygens' construction cannot explain it. Reflection, refraction and diffraction can all be explained using secondary wavelets.

(b) 0.25 ns

Speed in glass $v = \dfrac{c}{\mu} = \dfrac{3\times10^8}{1.5} = 2\times10^8$ m/s, so $t = \dfrac{d}{v} = \dfrac{0.05}{2\times10^8} = 2.5\times10^{-10}$ s $= 0.25$ ns.

(d) $BD/AC$

In the time $t$ that the point $B$ of the wavefront travels the distance $BD$ in air, the point $A$ travels the distance $AC$ in glass. So $BD = ct$ and $AC = vt$, and

$$n = \frac{c}{v} = \frac{BD}{AC}$$