Single-slit diffraction

Interference does not strictly require two sources. Even a single sufficiently narrow slit produces a pattern of maxima and minima on the screen: this is diffraction. The mechanism is explained by the Huygens–Fresnel principle: every point of the slit behaves as a small source of secondary waves, and these waves, coming from different points of the same aperture, interfere with one another on the screen. In certain directions they reinforce, in others they cancel.

Key formula — Single-slit minima

For a wave of wavelength $\lambda$ passing through a slit of width $a$, the intensity minima on the screen occur at angles: $\ev{a\,\sin\theta_k = k\,\lambda \qquad k = \pm 1, \pm 2, \pm 3, \dots}$

Note that the condition $a\sin\theta = k\lambda$ identifies the minima, not the maxima: it is the opposite of what happens in the double slit. The central maximum ($k = 0$) is the brightest and also the widest: its angular extent between the first two minima is $2\lambda/a$. The side maxima that follow are much weaker and progressively less intense.

Note

The width of the central maximum, $2\lambda/a$, tells us that a narrow slit (small $a$) diffracts strongly: the wave spreads out considerably after the aperture. A slit wide compared with $\lambda$ diffracts little and the light carries on almost in a straight line. Diffraction is noticeable only when the aperture is comparable with the wavelength.

Links

Topics: Onde Concepts: Diffrazione

Related exercises: Problema — Ecografia lunghezza d’onda degli ultrasuoni · Elettrone a 300 V e diffrazione cristallina · de Broglie di un elettrone a 70 V