Single-Slit Diffraction Minima

The equation looks exactly like the grating equation and means the opposite: here mλ = w sin θ locates the dark fringes, not the bright ones. The reason is that a single slit is not two sources but a continuous strip of them. Divide the slit in half; when the path difference between the two halves' matching points is exactly half a wavelength, every ray from the top half is cancelled by a partner from the bottom half, and the whole strip extinguishes itself. That condition works out to w sin θ = λ, then again in quarters for 2λ, in sixths for 3λ, and so on — which is why the orders count 1, 2, 3 and m = 0 is missing. There is no zeroth minimum, because at dead centre every ray arrives in step and the pattern is at its brightest.

The width of that central band is what the formula is usually asked for. A 0.05 mm slit in 600 nm light puts the first minimum at sin θ = 6 × 10⁻⁷/5 × 10⁻⁵ = 0.012, about 0.69°, so on a screen 2 m away the bright centre is roughly 4.8 cm across — twice the spacing of every band beyond it. Narrow the slit and the pattern spreads: the more tightly you confine the light, the more it insists on going sideways, which is diffraction's whole personality and the classical shadow of the uncertainty principle. Push the other way and it disappears — at w ≫ λ the first minimum crowds into the axis and the slit simply casts a shadow, which is why doorways diffract sound you can hear and not light you can see.