Physics 2102 Jonathan Dowling Lecture 28 Ch 35 Interference Interference Example A red light beam with wavelength 0 625 m travels through glass n 1 46 a distance of 1mm A second beam parallel to the first one and originally in phase with it travels the same distance through sapphire n 1 77 How many wavelengths are there of each beam inside the material In glass g 0 625 m 1 46 0 428 m and Ng D g 2336 45 In sapphire s 0 625 m 1 77 0 353 m UV and Ns D s 2832 86 What is the phase difference in the beams when they come out The difference in wavelengths is Ns Ng 496 41 Each wavelength is 360o so N 496 41 means Nx360o 0 41x360o 148o How thick should the glass be so that the beams are exactly out of phase at the exit destructive interference N D s D g D n2 n1 0 31 D m 1 2 A thickness D m 0 5 2 02 m would make the waves OUT of phase For example 1 008 mm makes them in phase and 1 010 mm makes them OUT of phase Thin Film Interference The patterns of colors that one sees in oil slicks on water or in bubbles is produced by interference of the light bouncing off the two sides of the film To understand this we need to discuss the phase changes that occur when a wave moves from one medium to the another where the speed is different This can be understood with a mechanical analogy Reflection Refraction and Changes of Phase Consider an UP pulse moving in a rope that reaches a juncture with another rope of different density A reflected pulse is generated The reflected pulse is also UP if the speed of propagation in the rope of the right is faster than on the left Low impedance The reflected pulse is DOWN if the speed of propagation in the right is slower than on the left High impedance The extreme case of ZERO speed on the right corresponds to a rope anchored to a wall Highest impedance If we have a wave instead of a pulse DOWN means 180 degrees OUT of phase and UP means 360 or IN PHASE Thin Films First reflected light ray comes from first interface second from second These rays interfere with each other n1 n2 n3 How they interfere will depend on the relative indices of refraction In the example above the first ray suffers a 180 degree phase change 1 2 a wavelength upon reflection The second ray does not change phase in reflection but has to travel a longer distance to come back up The distance is twice the thickness of the layer of oil For constructive interference the distance 2L must therefore be a half integer multiple of the wavelength i e 0 5 1 5 0 5 2n odd number In phase 2 L Anti phase 2 L integer 2 n2 n2 Thin Films Soap Bubbles If the film is very thin then the interference is totally dominated by the 180 phase shift in the reflection At the top the film is thinnest due to gravity it lumps at the bottom so one sees thefilm dark at the top 180 Air n 1 0 Soap n 1 Air This film is illuminated with white light therefore we see fringes of different colors corresponding to the various constructive interferences of the individual components of the white light which change as we go down The thickness increases steeply as we go down which makes the width of the fringes become narrower and narrower Reflective Coatings To make mirrors that reflects light of only a given wavelength a coating of a specific thickness is used so that there is constructive interference of the given wavelength Materials of different index of refraction are used most commonly MgFe 2 n 1 38 and CeO2 n 2 35 and are called dielectric films What thickness is necessary for reflecting IR light with 1064nm n 2 35 n 1 38 First ray 180deg Second ray 2L 2 n L Ceo2 n nm Third ray If wafer has the same thickness and is of the same material 4L 2 n 2 destructive Choose MgFe2 wafer so that 2n1L1 2n2L2 2 2n2L2 2 3 L2 n2 386 nm We can add more layers to keep reflecting the light until no light is transmitted all the light is either absorbed or reflected Anti Reflective Coatings Semiconductors such a silicon are used to build solar cells They are coated with a transparent thin film whose index of refraction is 1 45 in order to minimize reflected light If the index of refraction of silicon is 3 5 what is the minimum width of the coating that will produce the least reflection at a wavelength of 552nm n 1 45 Both rays undergo 180 phase changes at reflection therefore for destructive interference no reflection the distance travelled twice the thickness should be equal to half a wavelength in the coating 2L L 95 1nm n Anti Reflective Coatings Radar waves have a wavelength of 3cm Suppose the plane is made of metal speed of propagation 0 n is infinite and Stealth Fighter reflection on the polymer metal surface therefore has a 180 degree phase change The polymer has n 1 5 Same calculation as in previous example gives 3cm L 0 5cm 4n 4 1 5 On the other hand if one coated a plane with the same polymer for instance to prevent rust and for safety reasons wanted to maximize radar visibility reflective coating one would have 3cm L 1cm 2n 2 1 5 Michelson Interferometers As we saw in the previous example interference is a spectacular way of measuring small distances like the thickness of a soap bubble since we are able to resolve distances of the order of the wavelength of the light for instance for yellow light we are talking about 0 5 of a millionth of a meter 500nm This has therefore technological applications In the Michelson interferometer light from a source at the left in the picture hits a semiplated mirror Half of it goes through to the right and half goes upwards The two halves are bounced back towards the half plated mirror interfere and the interference can be seen by the observer at the bottom The observer will see light if the two distances travelled d1 and d2 are equal and will see darkness if they differ by half a wavelength Michelson Morley Experiment Michelson won the Nobel prize in 1907 for his optical precision instruments and the spectroscopic and metrological investigations carried out with their aid The interpretation of these results is that there is no displacement of the interference bands The result of the hypothesis of a stationary ether is thus shown to be incorrect A A Michelson Am J Sci 122 120 1881 The largest Michelson interferometer in the world is in Livingston LA in LSU owned land it is operated by a project funded by the National Science Foundation run by Caltech and MIT …
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