Diffraction Deriving nl = d sin q

Diffraction Diffraction results from the spreading out of waves passing through a gap. The narrower the gap, the more the waves spread out. If the gap is less than one wavelength, there is no diffraction at all. Wavelength is not affected by diffraction. Diffraction happens with all waves. Single slit and multiple slit diffraction have nothing to do with Young’s double slits. So don’t use the double slit formula. Saturday, 01 December 2018 Leeds City College

Single Slit Diffraction For a wide gap, there is not much diffraction. Light rays Sharp shadows Single bright region If the gap is made narrow, then we get to see a diffraction pattern Light rays Bright regions Central bright region Monochromatic light is needed for a clear pattern. Saturday, 01 December 2018 Leeds City College

Intensity and diffraction The observed pattern of bright and dark fringes can be explained by the superposition of secondary wavelets. The intensity is greatest in the middle. The bright fringes either side are much less intense. The further we move away from the centre, the dimmer they are. Saturday, 01 December 2018 Leeds City College

Equation Wavelength (m) Angle of spread. Light rays Bright regions Central bright region Wavelength (m) Angle of spread. If a is less than l, sin q > 1. That simply will not do. Wavelength of light are in nano-metres (nm). Convert to m. 1 nm = 1 × 10-9 m Width of gap (m) Saturday, 01 December 2018 Leeds City College

A further equation From the diagram we can see: D W Red light q For very small angles, tan q = sin q Since: Rearranging: we can write: Saturday, 01 December 2018 Leeds City College

Linking with Double Slits Fringe spacing Width of the central bright fringe Saturday, 01 December 2018 Leeds City College

Resolution Consider two objects very close together: Not resolved Just resolved Easily resolved This was first worked out by John William Strutt, 3rd Lord Rayliegh. It is often called Rayliegh’s criterion Saturday, 01 December 2018 Leeds City College

Diffraction Grating There are two kinds of diffraction grating: Transmission (where the light goes through); Reflection (where the light is reflected). The diffraction grating has the advantage over the double slit method of measuring wavelength in that: the maxima are more sharply defined the beam passes through more slits than two, so the intensity is brighter the angles are larger so that they can be measured with greater precision. Saturday, 01 December 2018 Leeds City College

How does diffraction work? q d Monochromatic light, wavelength l These rays go to the first bright fringe. Monochromatic light is shone onto a transmission grating which has N lines per metre. The slit spacing is given by: Bright spots appear only when there is constructive interference, i.e. when light from A is in phase with B. That happens when they are one whole wavelength apart. Saturday, 01 December 2018 Leeds City College

Deriving the equation d q Let’s look in close up: AC = 1 l Monochromatic light, wavelength l A B C q AC = 1 l AB = d AC = d sin q So we can write: Therefore: This is true for any of the bright spots, 1, 2, 3, …n Saturday, 01 December 2018 Leeds City College

Be careful! The diffraction grating is often given as lines per millimetre. You will have to convert to lines per metre. 750 lines per mm = 750 × 103 lines per metre. d = 1.33 × 10-6 m. In some gratings, the situation is even worse – they write N in lines per inch! (US–sourced gratings in particular. 10000 lines per inch = 393 per millimetre. The maximum order of diffraction possible is limited by sin q < 1. If you find that n = 3.8, then the maximum orders possible is 3, NOT 4. Wavelengths are in nanometres (nm) 1 nm = 1 × 10-9 m Saturday, 01 December 2018 Leeds City College

Patterns of Diffraction If the wavelength is reduced, we observe that the angle of each order is reduced. If we have white light, we will see several coloured lines for each order. Saturday, 01 December 2018 Leeds City College