Intensity distribution in interference

Interference is the modification of intensity produced when two or more coherent light waves superpose, giving maxima and minima in the resultant intensity.

Intensity distribution

 S1 o--------\
              \--> P   path difference Δ
 S2 o--------/

1. Let two coherent waves of amplitudes A₁ and A₂ and phase difference δ reach point P: y₁ = A₁ sin ωt, y₂ = A₂ sin(ωt + δ)
2. By the principle of superposition: y = y₁ + y₂ = A₁ sin ωt + A₂ sin(ωt + δ)
3. Expand: y = sin ωt (A₁ + A₂ cos δ) + cos ωt (A₂ sin δ)
4. Put A₁ + A₂ cos δ = A cos φ and A₂ sin δ = A sin φ, so y = A sin(ωt + φ).
5. Squaring and adding:

A² = (A₁ + A₂ cos δ)² + (A₂ sin δ)²
   = A₁² + A₂² + 2 A₁A₂ cos δ

6. Intensity is proportional to the square of the amplitude, so

I = A₁² + A₂² + 2 A₁A₂ cos δ

Maxima and minima

• Maximum: cos δ = 1, δ = 2nπ (path difference nλ): I_max = (A₁ + A₂)²
• Minimum: cos δ = −1, δ = (2n+1)π (path difference (2n+1)λ/2): I_min = (A₁ − A₂)²

If A₁ = A₂ = a: I = 4a² cos²(δ/2), varying between 4a² and 0. The average intensity is 2a², equal to the sum of separate intensities, so energy is conserved: it is only redistributed.