Interference: Double-Slit and Thin Films
Young's double-slit experiment is the single most direct proof that light behaves as a wave: two slits act as two coherent sources, and wherever the path lengths from each slit to a screen differ by a whole number of wavelengths, the waves arrive in phase and add constructively (bright fringe); a half-wavelength difference gives destructive interference (dark fringe). Thin-film interference is the same idea applied to reflections off the top and bottom of a thin coating, complicated by one subtlety: reflecting off a higher-index medium flips a wave's phase by half a cycle, which a naive path-difference calculation alone would miss.
What you'll learn
- Calculate the positions of bright and dark fringes in a double-slit interference pattern.
- Relate fringe spacing to slit separation, wavelength, and screen distance.
- Determine the path-length difference condition for constructive and destructive interference.
- Calculate the minimum thickness for constructive or destructive interference in a thin film.
- Correctly account for the half-wavelength phase shift that occurs on reflection off a higher-index medium.
- Explain why interference patterns require coherent light sources.
1. Young's Double-Slit Experiment
Two narrow, closely-spaced slits illuminated by a single coherent source act as two synchronized point sources. At any point on a distant screen, the two paths from the slits generally have different lengths — where that path difference is a whole number of wavelengths, the waves arrive crest-to-crest and add constructively (a bright fringe); where it's a half-integer number of wavelengths, they arrive crest-to-trough and cancel (a dark fringe).
For small angles, this translates into a simple, evenly-spaced fringe pattern on the screen: Δy=λL/d.
2. Thin-Film Interference
Light reflecting off a thin transparent film (a soap bubble, an oil slick, an anti-reflective coating) partially reflects at the top surface and partially at the bottom, and these two reflected beams interfere. The extra path length the bottom-reflecting beam travels is 2t (down and back up through the film, of thickness t and refractive index n).
The subtlety: reflecting off a boundary where the index *increases* flips the wave's phase by half a cycle; reflecting off a boundary where it *decreases* does not. Whether the top and bottom reflections both shift, neither shifts, or only one shifts determines which of 2nt=mλ or 2nt=(m+½)λ actually corresponds to constructive interference in a given setup.
Key equations
- d sinθ = mλ (bright), d sinθ = (m+½)λ (dark) — The angular condition for bright and dark fringes in double-slit interference.
- Δy = λL/d — Fringe spacing on a distant screen, valid for small angles — the distance between adjacent bright (or adjacent dark) fringes.
- 2nt = mλ or (m+½)λ — Path-length condition for thin-film interference — which side of the equation (mλ or (m+½)λ) corresponds to constructive vs. destructive depends on how many of the two reflections (top and bottom surface) involve a phase-flipping higher-index reflection.