Derive law of reflection of light …

Law of Refraction using Huygens' Principle:
Similarly, we can use a Huygens construction to illustrate the law of refraction.
Here we must take into account a different speed of light in the upper and lower media. If the speed of light in a vacuum is c, we express the speed in the upper medium by the ratio ni​c​, where ni​ is the refractive index. Similarly, the speed of light in the lower medium is nt​c​. The points D, E and F on the incident wavefront arrive at points D, J and I of the plane interface XY at different times. In the absence of the refracting surface, the wavefront GI is formed at the instant ray DF reaches I. During the progress of ray CF from F to I in time t, however , the ray AD has entered the lower medium, where the speed in different. Thus if the distance DG is vi​t, a wavelet of radius vt​t is constructed with center at D. The radius DM can also be expressed as  
DM=v_i​t 
=v_i​(​DG/v_i​)

=(​n₁/n_t​​)DG 

Similarly, a wavelet of radius n₁/n_t​​n₁/n_t​​ JH is drawn centered at J. The new wavefront KI includes a point I on the interface and is tangent to the two wavelets at points M and N. The geometric relationship between the angles i and t, formed by the representative incident ray AD and refracted ray DL, is Snell's law , which may be expressed as ni​sinθi​=ni​sinθi​.

Reflection of a plane wavefront of light at a plane surface

Where MN: Plane mirror,
RA and QC: Incident rays, 
AP: Normal to MN,
AB: Incident wavefront,
i: Angle of incident,
CE: Reflected wavefront,
 r: Angle of reflection

When wavefront AB is incident on the mirror, at first, point A becomes a secondary source and emits secondary waves in the same medium. If T is the time taken by the incident wavefront to travel from B to C, then BC = vT. During this time, the secondary wave originating at A covers the same distance, so that the secondary spherical wavelet has a radius vT at time T.

To construct the reflected wavefront, a hemisphere of radius vT is drawn from point A. Draw a tangent EC to the secondary wavelet.

The arrow AE shows the direction of propagation of the reflected wave.

AP is the normal to MN at A,

∠RAP = i = angle of incidence and

∠PAE = r = angle of reflection

In Δ ABC and Δ AEC,

AE = BC and ∠ABC = ∠AEC = 90°

∴ Δ ABC and Δ AEC are congruent.

∴ ∠ACE = ∠BAC = i        .....(1)

 Also, as AE is perpendicular to CE and AP is perpendicular to AC,

∠ACE = ∠PAE = r       .....(2)

∴ From Eqs (1) and (2),

i = r

Thus, the angle of incidence is equal to the angle of reflection. This is the first law of reflection. Also, it can be seen from the figure that the incident ray and reflected ray lie on the opposite sides of the normal to the reflecting surface at the point of incidence and all of them lie in the same plane. This is the second law of reflection. Thus, the laws of reflection of light can be deduced by Huygens' construction of a plane wavefront.