Reflection and Refraction of Plane Wave at Plane Surface Using Huygens' Principle

Reflection Using Huygens' Principle:

• Reflection is a sudden change in direction of propagation of a wave that strikes a boundary between two different media.
• If incoming rays are plane waves with infinite parallel planes, the wave AB falls on a reflecting surface and is incident perpendicular to the incident ray at angle i.
• AA₁ = BE (equal distances), triangles are congruent → ∠i = ∠r (angle of incidence = angle of reflection). ✓
• First Law of Reflection: ∠i = ∠r
• Second Law: Incident wavefront, reflected wavefront, and normal all lie in same plane perpendicular to reflecting surface.

Refraction Using Huygens' Principle:

• Refraction is the change in velocity of light as it passes from one medium to another.
• If a plane wavefront AB is incident on a surface, v₁ and v₂ are velocities in medium 1 and 2 respectively (v₁ > v₂).
• From Huygens' principle, A and C form the source of secondary spherical wavelets; time = t.

△ADC gives: \[\frac{\sin i}{\sin r}=\frac{\left(\frac{BC}{AC}\right)}{\left(\frac{AD}{AC}\right)}=\frac{BC}{AD}=\frac{v_{1}t}{v_{2}t}=\frac{v_{1}}{v_{2}}=\mu,\]

This proves Snell's Law of Refraction: \[\frac{\sin i}{\sin r}=\frac{v_{1}}{v_{2}}=n\mathrm{(constant)}\].