Physics 2102 Gabriela Gonz lez Light is a wave When two beams of light combine we can have constructive or destructive interference http www colorado edu physics 2000 applets fourier html The light travels more slowly in more dense media v c n n index of refraction Since the period T is the same the wavelength has to change v T Snell s law Wavelength Frequency 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 L g 2336 45 In sapphire s 0 625 m 1 77 0 353 m UV and Ns L 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 L s L g L n2 n1 0 31 L m 1 2 A thickness L m 0 5 2 02 m would make the waves OUT of phase For example 1 009 mm 499 5 x 2 02 m makes them come OUT of phase and 1 010 mm 500 0 x 2 02 m makes them IN 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 a transverse pulse moving in a rope that reaches a juncture with another rope of different density A reflected pulse is generated The reflected pulse is on the same side of the rope as the incident one if the speed of propagation in the rope of the right is faster than on the left The reflected pulse is on the opposite side of the rope if the speed of propagation in the right is slower than on the left The extreme case of no speed on the right corresponds to a rope anchored to a wall If we have a wave instead of a pulse being on the opposite side of the rope means 180 degrees out of phase or one half wavelength out of phase Phase with respect to incident beam r1 1 180o if n2 n1 1 0o if n2 n1 r2 2 2n2L 180o if n3 n2 1 2n2L if n3 n2 if we have air oil water or air Constructive interference 2n2L 2m 1 Destructive interference 2n2L 2m 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 MgFe2 n 1 38 and CeO2 n 2 35 and are called dielectric films What thickness is necessary for reflecting IR light with 1064nm First ray 180deg Second ray 2L 2 n L Ceo2 4 n 4 113nm 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 2n2 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 n 2 35 n 1 38 All points in a wavefront serve as point sources of spherical secondary waves After a time t the new wavefront will be the tangent to all the resulting spherical waves Christian Huygens 1629 1695 Young s double slit experiment Young s double slit experiment Path difference L d sin Bright fringe L m d sin Dark fringe L m 1 2 d sin The intensity on the screen is I I0 4cos2 2 with 2 d sin 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 semi plated 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 Einstein s messengers einsteinsmessengers org 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 and LSU collaborates in the project http www ligo la caltech edu Mirrors are suspended with wires and will move detecting ripples in the gravitational field due to astronomical events http www amnh org sciencebulletins sid a f gravity 20041101 src l American Museum of Natural History Science Bulletins Gravity Making Waves Examples Ocean waves moving at a speed of 4 0m s are approaching a beach at an angle of 30o to the normal The water depth changes abruptly near the shore and the wave speed there drops to 3 0m s Close to the beach what is the direction of wave motion Refraction index is inversely related to wave speed n2 n1 v1 v2 4 3 Snell s law n2sin 2 n1sin 1 2 asin n1 n2 sin 1 asin 75sin 30o 22o Example Solar panels 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 …
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