Huygens Principle and Interference

Wavefront

The locus of all points in a medium oscillating in the same phase at a given instant.

Coherent Sources

Sources of waves that have the same frequency, constant phase difference, and preferably similar amplitudes.

Constructive Interference

The phenomenon where two waves superpose in phase, resulting in a resultant wave of greater amplitude and intensity.

Destructive Interference

The phenomenon where two waves superpose out of phase, resulting in a resultant wave of smaller (or zero) amplitude and intensity.

Fringe Width

The distance between the centers of two consecutive bright fringes or two consecutive dark fringes in an interference pattern.

Path difference for Constructive Interference

n = 0, 1, 2, ... (for bright fringes)

Δx = nλ

Path difference for Destructive Interference

n = 0, 1, 2, ... (for dark fringes)

Δx = (n + 1/2)λ

Relation between Path Difference and Phase Difference

Δφ is phase difference, Δx is path difference, λ is wavelength

Δφ = (2π/λ)Δx

Fringe Width (Young's Double Slit Experiment)

λ = wavelength, D = distance between slits and screen, d = distance between slits

β = λD/d

Resultant Intensity in Interference

I₁, I₂ are intensities of individual waves, Δφ is phase difference

I_resultant = I₁ + I₂ + 2√(I₁I₂)cos(Δφ)

Resultant Intensity for Equal Amplitudes (a₁=a₂=a)

I₀ is intensity due to single source

I_resultant = 4I₀cos²(Δφ/2)

Ratio of Maximum to Minimum Intensity

a₁, a₂ are amplitudes of individual waves

I_max / I_min = ((a₁ + a₂)/(a₁ - a₂))²