Huygens Principle Interference

10.1 Introduction — From Particles to Waves

For centuries, physicists argued about what light really is. In 1637, Descartes treated light as tiny material particles, and Newton later refined this into the corpuscular theory — small "corpuscles" travelling in straight lines. But as early as 1678, the Dutch scientist Christiaan Huygens proposed that light is a wave, much like ripples spreading on a pond.

Newton's reputation kept the particle picture alive until Thomas Young's 1801 double-slit experiment produced unmistakable interference fringes — a signature only waves can create. By 1873, James Clerk Maxwell unified electricity, magnetism and optics: light is an electromagnetic wave.

10.2 Huygens Principle

A wavefront is the locus of all points that oscillate in the same phase. For a stone dropped in water, the circular ripples are the wavefronts. For light, three shapes are common:

• Spherical — from a point source radiating equally in all directions.
• Cylindrical — from a line source such as a slit illuminated by a lamp.
• Plane — from a very distant source (e.g. the Sun), where the curvature is negligible.

Huygens' construction

Huygens' geometrical principle has two statements:

1. Every point on a given wavefront acts as a fresh secondary source emitting tiny spherical wavelets in the forward direction, each travelling with the speed of the wave in the medium.
2. The new wavefront, at a later instant \(t\), is the forward envelope (tangent surface) of all these secondary wavelets.

10.3 Reflection and Refraction Using Huygens Principle

Deriving the Law of Reflection

Consider a plane wavefront AB falling obliquely on a plane mirror. As B travels a distance \(v\tau\) to hit the mirror at \(C\), the wavelet from A expands into a hemisphere of radius \(v\tau\). The tangent from \(C\) touches this hemisphere at \(E\). Triangles \(ABC\) and \(AEC\) share the hypotenuse \(AC\) and both have perpendicular sides of length \(v\tau\); hence they are congruent. Therefore:

Deriving Snell's Law of Refraction

Let light cross from medium 1 (speed \(v_1\)) into medium 2 (speed \(v_2\)). While the wavelet at \(B\) travels \(BC=v_1\tau\) to reach the interface, the wavelet from \(A\) in the second medium has expanded a radius \(AE=v_2\tau\). From the two right-angled triangles sharing \(AC\):

Dividing:

This is Snell's law: \(n_1\sin i=n_2\sin r\). Two important facts:

• Frequency \(\nu\) is unchanged on refraction — it is fixed by the source.
• Wavelength and speed change: \(\lambda_2=\lambda_1(v_2/v_1)=\lambda_1/n\) (for light going from vacuum into a medium of refractive index \(n\)).

10.4 Coherent and Incoherent Addition of Waves

The superposition principle states: when two or more waves meet at a point, the resultant displacement equals the algebraic sum of the individual displacements.

Coherent Sources

Let two coherent waves of amplitudes \(a_1,a_2\) and intensities \(I_1\propto a_1^2,\ I_2\propto a_2^2\) meet with phase difference \(\phi\). The resulting intensity is:

• Constructive (\(\phi=0,2\pi,\dots\)): \(I_\text{max}=(\sqrt{I_1}+\sqrt{I_2})^2\).
• Destructive (\(\phi=\pi,3\pi,\dots\)): \(I_\text{min}=(\sqrt{I_1}-\sqrt{I_2})^2\).
• For equal intensities \(I_1=I_2=I_0\): \(I=4I_0\cos^2(\phi/2)\), with \(I_\text{max}=4I_0\) and \(I_\text{min}=0\).

Incoherent Sources

If phase varies rapidly and randomly (two separate bulbs, for instance), \(\cos\phi\) averages to zero and intensities simply add:

Worked Examples