Multiple Beam Interference - The Diffraction Grating

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***Multiple Beam Interference - The Diffraction Grating*** is the sixteenth and final lecture in the Waves and Optics section of PH1011. It covers the effect of using multiple slits in a diffraction grating instead of just two, and the effects that this has on the resulting diffraction pattern.

## Content

### Diffraction Gratings

In previous lectures, diffraction gratings with two slits have been covered. Maxima occur where ***d*sinθ *= mλ***, with an intensity proporional to ***cos2(ϕ/2)***. Using phasors, this is seen to have **-->-->** at maxima and **--><--** at minima.

Multiple beam interference operates under the same concept, merely with multiple slits - and therefore multiple incoming beams which will interfere with one another. If the distance between slits *d* is constant between all slits, then all values of θ will be equal - and therefore the light will combine constructively, with maxima still occurring where dsinθ = mλ. However the peaks produced if this is graphed are far sharper than those produced by two beam interference, and a number of much smaller peaks are produced intermediately (number of peaks between 2π = number of slits). These smaller peaks have a negligable value, but the sharpness of the large peaks are useful for separating out distinct wavelengths.

The phase difference in this case remains ***ϕ = 2πd*sin*θ/λ***, and if the slit separation is equal with two slits or a number then the maxima occur at the same angles - but whilst minima first occur in two slit diffraction systems at ϕ = 2π/2, multiple beam systems have their first 0 point when ***ϕ = 2π/n***, n being the number of slits.

### Intensity Distribution

The overall available intensity in the system is equal regardless of how many slits the diffraction grating has - algebraically this can be defined using ET = ΣEi = ΣEi combined with I = b<ET2. However, it can more eaily be defined through phasors - with two beam phasors an initial phase difference at angle ϕ would not produce too different a resultant angle, but with for example six beams, and therefore six phase differences, a far larger resultant angle is produced.

### Butterflies

Diffraction occurs in nature; it can cause iridescence in the examples of things like feathers, butterfly wings, bubbles and mother-of-pearl. In butterflies this has a distinct biological advantage - as the flapping of the butterfly's wings causes light to interact constructively and destructively in no real order, the butterfly appears to disappear periodically as its wings reflect light in a destructive manner. In the case of the blue morpho, its colour is entirely formed by interference - it has no physical characteristics which are blue themselves. On a microscopic level butterflies' wings have grooves, which act as a diffraction grating (with the light being reflected rather than coming through from the other side). The blue colour is produced as the grooves are ~200nm apart - half of the wavelength of blue light, which allows only blue from the incoming white light to be reflected to allow constructive interference.

## Summary

Multiple beam diffraction gratings operate in a similar way to two beam diffraction gratings, however with more oppertunity for different interference. This causes an intensity distribution graph with multiple but slimmer peaks, which in practice can be used to separating wavelengths with ***dsinθ = mλ***. The phase difference in these systems is ***ϕ = 2πd*sin*θ/λ***. Multiple beam interference occurs in nature alongside conventional physics.