How Single-Slit Diffraction Minima Form

A finite opening makes waves spread

Diffraction is the spreading of a wave after it passes through an opening or around an obstacle. It becomes prominent when the opening is not enormously larger than the wavelength.

For a uniformly illuminated single slit, different points across the opening act like coherent secondary wave sources. At certain viewing angles, their contributions cancel in pairs. In the far field, these dark minima satisfy

$$a\sin\theta_m=m\lambda, \qquad m=1,2,3,\ldots,$$

where $a$ is slit width, $\lambda$ is wavelength, and $\theta_m$ is measured from the central axis. The central bright maximum lies between the $m=-1$ and $m=+1$ minima, so it is wider than the side maxima.

Fresh example

Light of wavelength $600\,\mathrm{nm}$ passes through a $3.00\,\mu\mathrm{m}$ slit. For the first minimum,

$$\sin\theta_1=\frac{600\times10^{-9}}{3.00\times10^{-6}}=0.200,$$

so $\theta_1=11.5^\circ$. For the second minimum,

$$\sin\theta_2=\frac{2(600\times10^{-9})}{3.00\times10^{-6}}=0.400,$$

so $\theta_2=23.6^\circ$.

Read the proportionality before calculating

• A narrower slit increases $\lambda/a$, pushing minima to larger angles and broadening the pattern.
• A longer wavelength also produces larger diffraction angles.
• If $m\lambda/a>1$, that order cannot exist because no real angle has $\sin\theta>1$.

Single-slit diffraction is an interference effect within one finite opening. It differs from the evenly spaced ideal two-source fringes of a narrow double slit, although real double slits show both interference and a single-slit diffraction envelope.