Diffraction and interference

Superposition and path difference

When two waves overlap, the total displacement is the sum of the individual displacements (the principle of superposition). If two waves of the same wavelength arrive in step (crest on crest) they reinforce: constructive interference, a bright point. If they arrive exactly out of step (crest on trough) they cancel: destructive interference, a dark point.

Whether two waves from two sources arrive in step depends on the path difference, the difference in the distances they travel:

• Constructive (bright): path difference $= m\lambda$, where $m = 0, 1, 2, \dots$
• Destructive (dark): path difference $= (m + \tfrac{1}{2})\lambda$

For a steady pattern the two sources must be coherent: the same frequency and a constant phase difference. Two separate light globes are not coherent. Young's trick was to split light from a single source into two using two slits, so the two parts stayed in step.

Young's double-slit experiment

Light passes through two very narrow slits a distance $d$ apart. Each slit diffracts the light, the two spreading beams overlap, and a screen shows a row of evenly spaced bright and dark bands called fringes.

When the screen is far away compared with $d$, the two rays heading towards a point on the screen are nearly parallel, both at angle $\theta$ to the centre line. The extra distance travelled by the ray from the lower slit is $d\sin\theta$. Bright fringes occur where this is a whole number of wavelengths:

$$d \sin\theta = m\lambda$$

$m$ is the order of the fringe. The central bright fringe ($m = 0$) has zero path difference. For small angles, $\sin\theta \approx \tan\theta = y/L$, where $y$ is the distance on the screen from the central fringe and $L$ is the distance to the screen. So the $m$th bright fringe is at $y = m\lambda L/d$ and the fringes are evenly spaced:

$$\Delta y = \frac{\lambda L}{d}$$

Wider fringes come from longer wavelengths, a more distant screen or slits closer together. Because $\lambda$ is tiny, $d$ must be small (a fraction of a millimetre) for the fringes to be visible. This was the first method to measure the wavelength of light.

Single-slit diffraction

Even one slit makes a pattern. Each point across the slit width $a$ acts as a Huygens wavelet source. Straight ahead, all wavelets arrive in step and give a bright central maximum. At the angle where $a\sin\theta = \lambda$, the light from the top half of the slit cancels the light from the bottom half point by point, giving the first dark band. Dark bands occur at

$$a \sin\theta = m\lambda, \qquad m = 1, 2, 3, \dots$$

Notice this looks like the double-slit equation, but here it gives dark bands. The central maximum is twice as wide as the others and much brighter. A narrower slit gives a wider pattern.

In a real double-slit experiment, each slit has a width too, so the double-slit fringes are bright only inside the single-slit pattern, called the envelope. Where a double-slit bright fringe falls exactly on a single-slit dark band, that fringe goes missing.

Diffraction gratings

A diffraction grating has hundreds or thousands of equally spaced slits per millimetre. The same condition, $d\sin\theta = m\lambda$, gives the bright directions, with $d$ the spacing between neighbouring slits. With many slits, light cancels almost everywhere except very close to those angles, so the maxima are very sharp and bright, and widely spaced because $d$ is so small.

Different wavelengths diffract to different angles (longer wavelengths further), so white light is spread into a spectrum in each order. That makes gratings ideal for spectroscopy: they give precise angles and so precise wavelengths. Astronomical spectrographs, including the ones used for the stellar spectra in 7D, use gratings.