Double-Slit Fringe Spacing

Illuminate two narrow slits with the same wave and follow the two paths to a point on a distant screen. The path lengths differ, and the difference grows as you move off the centre line. Where it works out to a whole number of wavelengths the two waves arrive in step and reinforce; where it is a half-odd number they arrive opposed and cancel. For small angles the path difference is $d\sin\theta \approx d\,y/L$, so bright fringes land at $y = m\lambda L/d$ and the spacing between neighbours is $\Delta y = \lambda L/d$. Notice what has dropped out: the order number $m$. The fringes are evenly spaced, and that even spacing is the signature that tells you at a glance you are looking at interference rather than at a shadow.

Green light at 500 nm through slits 0.25 mm apart, with the screen 1 m away, gives $\Delta y = (500 \times 10^{-9})(1)/(0.25 \times 10^{-3}) = 2.0$ mm. A wavelength of half a micrometre has produced a pattern you can measure with a school ruler, and the geometry is doing the amplification: $L/d$ here is a factor of 4000. Run the same relation backwards — measure $\Delta y$, $L$ and $d$, solve for $\lambda$ — and you have measured the wavelength of light on a desk with no special apparatus. That is precisely what makes the experiment famous.

Thomas Young presented this to the Royal Society in 1803, and the usual telling — that it settled the wave-versus-particle argument overnight — is not what happened. Young was attacked, hard, notably in the Edinburgh Review, and Newton's authority held in Britain for another fifteen years. What actually turned the profession was Augustin Fresnel's mathematical wave theory in 1818 and the episode around it: Siméon Poisson, judging the prize competition, pointed out that Fresnel's theory absurdly predicted a bright spot at the exact centre of a circular object's shadow, whereupon François Arago went and looked, and found it. One more detail worth knowing, because it is almost always misdescribed: Young's own arrangement was not a pair of slits cut in a card. He split a narrow sunbeam by holding a thin slip of card edge-on in it. The two-slit form is the modern classroom version of the idea.

Three cautions. This is a small-angle result, $\sin\theta \approx \tan\theta \approx \theta$ — and it is good to about 1% out to 10°, which covers essentially every real double-slit setup because $\lambda/d$ is small. Take it to a diffraction grating, where the angles run to tens of degrees, and it fails badly; that is exactly why the grating page carries the exact form instead. Second, $d$ is the separation between slit centres, not the gap between their edges and not the width of a slit. Slit width does matter, but it does something different: it imposes a broad single-slit diffraction envelope that modulates how bright each fringe is, and can extinguish some of them entirely wherever the ratio of separation to width is a whole number. The spacing is set by $d$ alone. Third, $\lambda$ must be the wavelength in whatever medium the light is crossing. Submerge the whole apparatus and the fringes crowd together by a factor of 1.33, because the wavelength shortens in water while the frequency does not change at all.