Calculate fringe spacing, wavelength and orders using the small-angle approximation.
Method and worked examples
In a double-slit arrangement, diffraction spreads light from each slit so the waves overlap. Coherent illumination gives bright fringes where the path difference dsinθ=mλ, with m=0 at the central maximum. Dark fringes lie halfway between successive bright fringes for equal slit amplitudes. The central bright fringe has equal path lengths from the two slits.
For a distant screen and small angles, sinθ≈tanθ≈Ly. Bright-fringe positions are ym=dmλL and adjacent bright-fringe spacing is Δy=dλL. Use metres consistently and distinguish a distance across several fringe intervals from one spacing. This model treats slit separation as much larger than each slit width, so the diffraction envelope varies more slowly than the fine interference fringes.
Example 1
Light of wavelength 500 nm passes through slits separated by 0.250mm onto a screen 2.00m away. Find fringe spacing.
Δy=dλL=500×10−9×0.250×10−32.00=0.00400m=4.00mm.
Example 2
Five bright-fringe intervals span 20.0mm. Find the adjacent spacing.
Δy=520.0=4.00mm. Count intervals, not just the number of bright marks.
Watch out:The small-angle formula has Δy ∝ d1, so larger separation produces closer fringes.