Wave superposition If two waves are in the same place at the same time they superpose. This means that their amplitudes add together vectorially Positively when they are in phase

Wave superposition If two waves are in the same place at the same time they superpose. This means that their amplitudes add together vectorially Negatively when they are in antiphase

The conditions for two waves to interfere with each other The waves must be coherent. This means there must be a constant phase difference between them ( which also implies that they have to be of the same frequency.) Interference between water waves from coherent sources

Interference Is best understood with relatively long waves Interference Is best understood with relatively long waves. Both water waves or microwaves (3cm ) provide a reasonable model. Coherence of the waves is ensured by obtaining the waves from a single frequency microwave source ( a monochromatic source) The waves are passed through two slits and are diffracted in the process. (That is they begin to spread out as if the slit was at the centre of a circular wave front) On a wave front diagram the waves positively superpose where they cross and negatively superpose in the centre of the gaps.

A microwave detector moved normal to the source detects positive and negative superposition called interference fringes.

Each peak is produced by positive interference Each peak is produced by positive interference. Each trough occurs because of negative interference

Central maximum phase difference 0 Here the path lengths from S1 and S2 to X are the same

path difference to next peak is 1 wavelength 2 wavelenghts S1 3 wavelengths S2 Note that when the phase difference is 2π the path length from one of the slits is longer by a single wavelength

Path difference 2 wavelengths Path difference 1 wvelenght Path difference 0

Measuring the difference in path length to the first fringe Zero or central fringe 0.670m In this case λ= 0.670-0.645= 0.025m.

Interference with Light Sources The geometry of the situation gives us the relationship W is the distance between adjacent fringes S is the slit separation D laser double slit The laser is a coherent light source which is divided into two by the fine double slit The screen is at a distance of 5-10m. The interference pattern below is produced. The fringes are equally spaced.

In reality you would measure the total distance between the centre of several visible fringes and divide by the number of dark intervals between them to achieve a better value for w.

Diffraction of Waves

Diffraction from a Single Slit Through a narrow single slit the wave front spreads out. If the slit is wide the spreading is slight. If the slit is comparable in width with the wavelegth of the wave the spreading is large.

Diffracted laser light from a single slit projected onto a screen Diffraction of water waves from above

intensity 2nd order Principle maximum 1st order Principle maximum Zero order principle maximum 1st order Principle maximum 2nd order Principle maximum minimum minimum minimum minimum

The diffraction grating A typical diffraction grating is an arrangement of identically spaced diffracting elements. Normally a large number of parallel lines are ruled on glass. The diffracting elements are the gaps between the ruled lines. typically there would be around 600 lines per mm.

Each line acts as a very narrow slit B θ θ C The light through each slit is diffracted in all directions. Consider the light through two slits diffracted at angle θ to the normal If the light diffracted through angle θ at A is in phase with the light diffracted through θ at B it must be in phase with the light at this angle through every other slit.

The path difference is the length AN θ A X N d θ Y Notice that: to be in phase the path difference (ie the distance A to N) has to be a multiple of the wavelength λ i.e (n λ ) The distance AN = d sinθ So d sinθ = n λ

What you see When the waves interfere constructively through each slit they are at an angle given by the formula d sinθ = n λ

Question When a grating of 300 lines per millimetre is illuminated with parallel beam of monochromatic light normal to it a second order principle maximum is observed at 18.90 to the straight through direction. Calculate the wavelength of the light.