Chapter 24 Wave Optics What is the color of gasoline Destructive Interference 24 1 Conditions for Interference Constructive Interference 1 1 1 t 90 1 1 1 In Phase t 1 180 270 360 Out of Phase t 180 degrees 2 2 t 2 t 1 t 2 This is how noise CANCELLING headphones work NOT how noise isolating headphones 1 Interference for Sound For example a pair of speakers driven in phase producing a tone of a single f and l l1 l2 Interference for Light Can t produce coherent light from separate sources it oscillates too fast for us to control f 1014 Hz Need two waves from single source taking two different paths Two different paths If l2 l1 l 2 there is no sound at all Interference possible here Single source Waves maintaining phase difference are considered to be coherent Sometime we are not good in identifying loudness accurately We do better if the sound changes continuously If two waves have different wavelengths they can t match all the time sometimes in phase and other times out of phase see graph Simulation 2 Distance is everything Away from the center in phase Out of phase 3 24 2 Young s Double Slit Light waves from a single source travel through 2 slits before meeting at the center on a screen The interference will be The reason for the dark spots is 1 Constructive destructive interference the two waves are out of phase one wave is shifted by half a wavelength relative to the other all above 2 Destructive 3 Depends on L d The rays start in phase and travel the same distance so they will arrive in phase Single source of monochromatic L It is BRIGHT at the light l center when there is no 2 slits line of sight Screen a distance separated by d L from slits At points where the difference in path length is 0 l 2l the screen is bright constructive d Single source of monochromatic light l 2 slitsseparated by d L At points where the difference in path l 3l 5l length is 2 2 2 the screen is dark destructive Screen a distance L from slits Mathematics d d Path length difference d sin Constructive interferencedsin ml 1 Destructive interference d sin m l 2 where m 0 or 1 or 2 4 A little geometry y L y d mlL d If is small sin tan y L Constructive interference dsin ml 1 Destructive interference d sin m l 2 mlL d 1 lL y m 2 d y where m 0 or 1 or 2 Example For a Young s double slit setup l 600 nm two slits are separated by 6 m and the screen is 1 0 m away What is the width of the 2nd order m 2 maximum y mlL d 1 lL y m 2 d 1 lL y m 2 d Combination of lenses surfaces etc One lens surface etc at a time The image of the first is the object of the second Mtotal M1 x M2 with all the signs included 5 24 3 Change of Phase Due to Reflection Flip or no flip n2 n1 n2 No flip when n1 n2 Reflected wave Incident wave n1 n1 n2 Upon reflection from a boundary between two transparent materials the phase of the reflected light may change Flip when n1 n2 If n1 n2 no phase change upon reflection If n1 n2 phase change of 180 upon reflection equivalent to the wave shifting by l 2 24 4 Thin Film Interference 1 1 2 t n1 1 0 air t Get two waves by reflection off of two different interfaces n1 1 0 air n2 1 5 thin film n2 n2 1 5 thin film n2 2 Distance Ray 2 travels approximately 2t further than ray 1 Phase change at interface for this particular case Two things that can change the phase Ray 1 180o phase change equivalent to l 2 shift 1 Traveling over some distance Ray 2 no phase change 2 Change of phase by 180 or not at interfaces The two have to be combined When looking at interference the two have to be combined 6 Constructive interference highly reflective 1 2 n 1 0 air t Ray 1 changes by l 2 upon reflection n1 thin film 1 2 n 1 0 air t n1 thin film 2nt ml m 1 2 3 Antireflection coating If Ray 2 comes back to the top surface in phase with Ray 1 The phase change due to traveling has to be ln 2 or ln 2 m ln m ln 2t m ln Destructive interference in this case no reflection The two equations above only work when one of the surfaces causes a change of phase by 180o If both surfaces or none cause such a shift the equations will be the opposite try to reason why ln l n Summary Or 2nt m l m 0 1 2 2nt ml 2nt m l 1 phase change destructive 0 2 phase change constructive constructive destructive Example A solar cell n 1 6 is typically coated with a film n1 1 3 to reduce reflection For a wavelength of 520 nm what is the smallest thickness of the film Color dependent 2nt m l 2nt ml 7 24 6 7 Diffraction Wall shadow For thin film interference it is constructive interference if ray 2 has to experience 1 no phase change 2 a total phase change l more than ray 1 3 the same total phase change as ray 1 bright This is not what is actually seen Screen with opening or obstacle without screen Diffraction Huygens Every point on a wave front acts as a source of tiny wavelets that move forward Central maximum 1st minima Light waves originating at different points within opening travel different distances to wall and can interfere We will see maxima and minima on the wall 8 Single Slit Diffraction sin m l a When the opening is circular m 1 2 3 a is the width of the slit THIS FORMULA LOCATES MINIMA dark bands or destructive interference Narrower slit broader pattern min 1 22 l a Most of the time everyday situations we have w l and 0 no observable diffraction y mlL d 1 lL y m 2 d All depend on l 24 9 Polarization of Light Waves 24 8 Diffraction Grating Grating multiple slits usually thousands or more dsin ml m 0 1 2 Maximum 9 Unpolarized Light on Linear Polarizer Linearly Polarized Light on Linear Polarizer Law of Malus TA Itransmitted Iincident cos2 is the angle between the incoming light s polarization and the transmission axis TA Most light comes from electrons accelerating in random directions and is unpolarized Averaging over all directions Stransmitted Sincident Incident E Transmission axis ETransmitted Eincidentcos Example Unpolarized light coming in Ii known If As light passes through a polarizer it will 1 be completely blocked by the polarizer 2 have to change its intensity 3 take on a polarization parallel to the transition axis 10