Examples of Two-Beam Interference

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***Examples of Two-Beam Interference*** is the 15th lecture in the Waves and Optics section of PH1011. It covers further two beam interference with the example of light waves in Michelson Interferometers and anti-reflective coatings.

## Content

### Coherence

* *Temporal coherence* refers to order in the direction of a wave's propogation (ie the same direction that the wave is travelling). It is therefore dependant on the frequencies/wavelengths of the waves, and without it the observed phase difference will vary with time. (Longitudinal)
* *Spatial coherence* refers to order in the direction of the wave front - ie which direction the waves are travelling. If waves propogate at different angles, they cannot form an intelligable interference pattern. (transverse)

Eg - waves propagating in the same direction but with differing wavelengths are spatially coherent but not temporally coherent. Waves of equal wavelength but differing directions of travel are spatially incoherent.

Interference patterns can only be observed when waves exhibit both temporal and spatial coherence.^

### Michelson Interferometers

Invented by Albert Michelson, the Michelson inferometer is a simple device used to split a coherent source of light (eg a wavelength stabilised laser) into two separate beams and then recombine them to extract information - for example the wavelength of the incident light. This is enabled by the initial beam being passed through a beam splitter - causing half of the light to pass directly though and the other half to be reflected at a right angle to prior propogation - and then being reflected back via a pair of mirrors, causing them to recombine upon passing the beam splitter again. The resulting single beam of light will now have a different form due to the slight phase difference caused by setting the mirrors to different distances - and this allows the manual changing of the interference to destructive or constructive. Moving one mirror half a wavelength's distance will cause a cycle from one maxima to the next (the beam being reflected doubles this distance, hence why it is not merely one wavelength); therefore if a number N of maxima are seen, then the distance moved is ***D = Nλ/2***. Given that the detector can be digital and therefore count accurately a great number of maxima, this allows for accurate measurement of very small displacements.

In itself, this set up is difficult to make experimentally sound as the mirrors must be set exactly perpendicular to the incoming and outgoing light beams. To allow for a more practical approach, corner cubes (triangles of glass) can be used to allow total internal reflection to negate the necessity of these angles being exact.

Doppler shift can also be used to describe this - as the mirror moves, different frequencies are produced (as although the observer is stationary, the apparent source of one beam appears to be moving). This is described as before by the relationship ***fob = fo(1± v/c)***.

### In Anti-reflection Coatings

Anti-reflective coatings appear on camera lenses, glasses, windows ect in order to reduce glare and improve vision through these lenses. They work simply by having a refractive index of the correct magnitude to cause reflected rays from the coating to interfere destructively with the rays reflected from the lens itself. In theory this only works with a given single wavelength of light; however in practice the coating can be set to destroy green light reflections and still be very effective (although this does cause a slight purple tint to the reflections that do exist, as blue and red light are not completely removed). The path difference between the two reflected beams must be ~2d; ideally *2d=λf/2*; as *λf=λo/nf* then ***d = λo/4nf***.

## Summary

* Temporal coherence and spatial coherence refer to order in the direction of wave propogation and fronts, respectively.
* Michelson Interferometers use a beam splitter and mirrors to allow interference patterns to form. This pattern can be used to accurately measure short distances, in the relationship **D = Nλ/2**.
* Doppler effect can be said to be acting within Michelson Interferometers.
* Anti-reflective coatings work by creating further reflections in order to destructively cancel initial reflections out.