Interference and diffraction

double slitYoung's experimentdiffraction gratingthin film interferenceoil slick colours

Where light waves reinforce and cancel — two slits, many slits, and the thin films that colour soap bubbles and oil.

Double-Slit Fringe Spacing

\Delta y = \frac{\lambda L}{d}

Spacing between adjacent bright fringes on a screen a distance L behind two slits d apart.

Diffraction Grating Equation

m \lambda = d \sin\theta

Bright beams leave a grating at angles where the path difference d sin θ is a whole number of wavelengths.

Thin-Film Constructive Interference (Bright Reflection)

2 n t = \left(m + \tfrac{1}{2}\right)\lambda

Bright-reflection condition for a thin film with one half-wave inversion: twice the optical thickness is a half-odd number of wavelengths.

Thin-Film Destructive Interference (Dark Reflection)

2 n t = m \lambda

Dark-reflection condition for a thin film with one half-wave inversion: twice the optical thickness is a whole number of wavelengths.

How they fit together

All four come from one idea: two paths of different length recombine, and what you see depends on whether the difference is a whole number of wavelengths or a half-integer number. Two slits give broad, evenly spaced fringes; thousands of slits in a grating give the same maxima far sharper, which is what makes gratings instruments rather than demonstrations. A thin film does it with one surface instead of many.

Thin films carry an extra term that catches almost everyone: reflection off a denser medium flips the wave by half a cycle, so the condition for a bright reflection picks up an extra half wavelength. That single inversion is the soap-bubble and oil-slick case. Coat glass with a film of intermediate index and both surfaces invert, the half-wave cancels, and the two conditions swap — which is exactly how anti-reflective lens coatings are designed.