Diffraction of Light

1. Introduction

Diffraction is the bending of waves around the edges of an obstacle or aperture and their spreading into the region of the geometrical shadow. It is a property of all waves (sound, water waves, light, X-rays) and is direct evidence of the wave nature of light.

Condition for appreciable diffraction: diffraction is significant only when the size of the obstacle or aperture (slit width $a$) is comparable to the wavelength $\lambda$:

• If $a \gg \lambda$: negligible diffraction, light travels in nearly straight lines (sharp geometrical shadow).
• If $a \approx \lambda$: strong diffraction, the waves spread widely.
• If $a < \lambda$: the wave spreads out in almost all directions.

This is why sound diffracts easily around doors and buildings (its wavelength is a few centimetres to metres, comparable to the openings), while light does not noticeably bend around ordinary objects (its wavelength is about $500\ \text{nm}$, far smaller than everyday obstacles).

Difference between interference and diffraction:

2. Diffraction at a Single Slit

When monochromatic light of wavelength $\lambda$ passes through a single slit of width $a$ and falls on a distant screen, the pattern consists of:

• a broad, bright central maximum centred on the straight-through direction,
• secondary maxima on both sides, each dimmer and narrower than the central one,
• dark fringes (minima) separating the maxima.

Condition for minima (dark fringes). By Huygens' principle every point of the wavefront in the slit is a secondary source. Consider the slit divided into two halves. Light from the top edge and from the middle of the slit travel to a point on the screen at angle $\theta$ with a path difference of $\dfrac{a}{2}\sin\theta$. The two halves cancel in pairs (destructive interference) when this equals $\dfrac{\lambda}{2}$, i.e. when $a\sin\theta = \lambda$. Repeating the argument for the slit divided into $2n$ strips gives the general condition for the $n$th minimum:

$\boxed{a\sin\theta_n = n\lambda}, \qquad n = \pm 1, \pm 2, \pm 3, \dots$

Important exam point: for a single slit, $a\sin\theta = n\lambda$ gives the minima (dark fringes), whereas in the double-slit and grating cases $d\sin\theta = n\lambda$ gives the maxima. This is a common source of confusion.

Width of the central maximum. The central maximum lies between the first minima on either side ($n = \pm 1$). For small angles, $\sin\theta \approx \theta \approx \lambda/a$, so:

• Angular width of central maximum: $\ \dfrac{2\lambda}{a}$
• Linear width on a screen at distance $D$: $\ \dfrac{2\lambda D}{a}$

Effect of narrowing the slit: as $a$ decreases, $2\lambda/a$ increases, so a narrower slit gives a wider central maximum. This is counter-intuitive but correct: the smaller the aperture, the more the light spreads.