Waves and Polaization in Geneal Wave means a distubance in a medium that tavels. Fo light, the medium is the electomagnetic field, which can exist in vacuum. The tavel pat defines a diection. The distubance can also have a diection (o none). The diection of tavel and the diection of distubance don t have to be the same. Fo example, conside a Slinky sping. If I have a Slinky stetched out in the z diection, I can launch waves down it by wiggling it in the x diection. I can also wiggle it in the y diection. The wave is tavelling in z, but the distubance is in the x o y diection. I could also wiggle the end of the sping in the z diection. That will make density waves tavel down the sping. These diffeent wiggles ae diffeent polaizations. Thee ae two tansvese polaizations and one longitudinal polaization.
Waves and Polaization (2) Of couse, I could wiggle the Slinky, o a ope, stetched in the z diection, in a plane at any angle tansvese to the z diection, and still geneate a wave that would tavel down the ope. I m not limited to x and y diections. The tansvese polaization is a vecto, it has diection as well as magnitude. I can epesent that vecto by its x and y components. Of couse, I could have chosen a diffeent coodinate system, with x and y axes that ae otated elative to the x and y axes (but with the same z axis). Then I would wite the vecto with numbes fo its x and y components that ae diffeent fom the numbes fo the x and y components, even though they epesent the same physical vecto. This concept that we can decompose the polaization of the same physical wave into diffeent components when we use diffeent coodinate systems is essential to undestand what happens with polaizing filtes.
Polaizing Filtes Imagine I have a ope to launch waves down, and at some distance away the ope goes though a fictionless slot just baely big enough fo the ope. If I wiggle the ope paallel to the slot, the ope is fee to slide in the slot, and the wave gets though. If I wiggle the ope pependicula to the slot, the ope can t wiggle in the slot, so the wave doesn t get though (it will eflect back to me o get absobed).
Polaizing Filtes (2) If I wiggle the ope at 45, what happens? The ope slides in the diection that the slot allows, so a wave does come out the othe side, but at lowe amplitude Also the polaization is paallel to the slot athe than to the initial 45 diection. The same thing happens with light and polaizing filtes. The amount of light that gets though the filte depends on how paallel the polaization of the light is to the filte. If the polaization is paallel, it nealy all gets though. If the polaization is at ight angles, none gets though. But what about the geneal case, whee the angle between the light polaization and the filte is abitay?
Polaizing Filtes (3) To get a quantitative answe, the tick is to expess the incoming polaization vecto in a coodinate system lined up with the axis of the filte. If a vecto points in the x diection in one coodinate system, and anothe coodinate system is otated aound the z axis by angle θ, what ae the components of the same physical vecto in the new coodinate system? y y θ x θ V x The x component in the otated system is x = V cosθ = x cosθ The y component in the otated system is y = V sinθ = xsinθ
Polaizing Filtes (4) The electic field component in the wave field that is paallel to the filte axis is tansmitted, and the electic field component that is pependicula to the filte axis is not tansmitted. The tansmitted electic field stength is just cosθ times the oiginal field stength, whee θ is the angle between the oiginal field diection and the filte axis. Now cosθ is a function that can be positive o negative, and that s sensible when we ae talking about electic field vectos. The electic field that is positive in one coodinate system can be negative in anothe coodinate system. But the intensity of light can t be negative! The intensity (powe) of light tuns out to depend on the squae of the electic field. So the light intensity getting though the filte is popotional to cos 2 θ. This is often called Malus Law.
Two Polaizing Filtes If I stat with unpolaized light, what intensity will get though an ideal polaizing filte The value of cos 2 θ, aveaged ove all θ, is 1/2, so half the light will get though (with a eal filte, it will be somewhat less than half, because thee is some absoption of the ight polaization too). What if I have a second filte downsteam of the fist? Remembe, the light that gets though the fist filte is completely polaized. If the second filte is paallel to the fist, whateve gets though the fist slot will get though the second filte. If the second filte is at ight angles to the fist filte, nothing that gets though the fist filte will get though the second. If the angle is in between, a faction will get though. The faction will be in fact cos 2 θ. What happens if I have two filtes, cossed at ight angles so no light gets though, and I put anothe filte between?
Electomagnetic Waves and Polaization Light is an electomagnetic wave. The electic and magnetic fields of the wave ae always at ight angles to the diection of popagation, and to each othe. The diection of the electic field is the polaization diection. One paticula wave solution to Maxwell s Equations is E = E 0 x ˆ cos( kz ωt) B = E 0 y c ˆ cos kz ωt ( ) ω k = c The solution above is linealy polaized in the x diection. The convention is that the polaization diection is the diection of the electic field. A wave moving in the z diection linealy polaized in the y diection would look like E = E 0 y ˆ cos( kz ωt) B = E 0 x c ˆ cos kz ωt ( ) ω k = c We can supepose both of these fields to get a wave polaized at +45 : E = E 0 ( x ˆ + y ˆ )cos( kz ωt) B = E 0 ( y ˆ x ˆ )cos( kz ωt) c A wave polaized at -45 is physically totally diffeent but looks vey simila mathematically: E = E 0 ( x ˆ y ˆ )cos( kz ωt) B = E 0 ( y ˆ + x ˆ )cos( kz ωt) c
Polaization and Speed of Light Fo vacuum, o ai, o wate, o glass, the speed of light is the same fo any polaization (although it may depend on fequency o wavelength). If a mateial has some intenal stuctue that makes the electical popeties diffeent in some diections, the speed of light can be diffeent fo diffeent polaizations. This is called dichoism. Many cystals ae like this. Quatz is an example. It has the same chemical makeup as glass (silicon dioxide), but glass has andom bond diections and oientation, while quatz is vey egula. If the polaization of light is paallel to some diections, the speed of light is diffeent. Mateials that ae stetched can also have this popety. Fo instance, tanspaency plastic is stetched duing manufactuing, so the speed of light is diffeent fo polaization paallel and at ight angles to the stetch axis. Watch what happens when I put this tanspaency between two cossed polaizes!
Polaization and Speed of Light (2) The stetching means the fast and slow polaization axes ae paallel and pependicula to the vetical axis, 90 degees apat. When the tanspaency is vetical o hoizontal, the light that gets though the fist polaize is aligned with eithe the fast o slow axis, and popagates at whateve speed that implies. The light gets blocked by the second polaize. When the tanspaency is at 45 degees, we can decompose it into components paallel to the fast axis, and paallel to the slow axis. The two components will have equal amplitudes. Initially, they ae in phase with each othe. But they tavel at diffeent speeds, so they get out of phase. Even though the amplitudes of the two 45-degee components ae still the same, because they got out of phase, they no longe add up to the oiginal light with the oiginal polaization. Since some of the light has the wong polaization, it gets though the second polaize.
Cicula Polaization Going back to ou stetched ope analogy, we aen t limited to shaking the end of the ope back and foth in a single plane. We could also move the end in a cicle. That would poduce a helical wave tavelling down the stetched ope. The motion would not be confined to a single plane. This is cicula polaization (the kind of polaization we have been talking about befoe is linea polaization). The motion of the end of the ope would be descibed as x = Rcos ±2πft y = Rsin ±2πft ( ) = Rcos ( ±ωt) = Rcos( ωt) ( ) = Rsin ( ±ωt) = ±Rsin( ωt) The fequency in cycles pe second is f, the fequency in adians pe second is ω. The plus-minus signs ae because we could be going clockwise o counteclockwise. The distubance would tavel down the ope in the z diection as waves. The displacement at othe points is ( ) = Rcos( kz ωt) ( ) = ±Rsin( kz ωt) x z y z Basically, we have waves in both x and y, out of phase in time and space.
Cicula Polaization of Light We can also make ciculaly-polaized light. To do that, we add x and y linea polaizations, but we also make them out of phase in time. A ight-cicula polaized wave looks like E = E 0 [ x ˆ cos( kz ωt) + y ˆ sin( kz ωt) ] B = E 0 y ˆ cos( kz ωt) x ˆ sin( kz ωt) c [ ] The electic field at a given point in space otates in diection at constant magnitude. The magnetic field otates the same diection but is always 90 away. A left-cicula wave looks like E = E 0 [ x ˆ cos( kz ωt) y ˆ sin( kz ωt )] B = E 0 y ˆ cos( kz ωt)+ x ˆ sin( kz ωt) c [ ] This sounds petty delicate, but some systems do it natually (some micowave antennas, some atomic tansitions).
Linea To Cicula Filte We can make a device that will tun linealy polaized light to cicula polaized light and back again. Get a dichoic mateial with fast and slow axes that ae 90 degees apat (in diection). Aange it so the fast and slow axes ae at +/- 45 degees elative to the linea polaization. Make the laye just the ight thickness such that the fast and slow waves will be 90 degees out of phase in time and space. The light that comes out will be ciculaly polaized. What happens if we make the laye thick enough that the slow wave is exactly 360 degees out of phase with the fast wave? What happens if we make the laye thick enough that the slow wave is exactly 270 degees out of phase with the fast wave? What happens if we make the laye thick enough that the slow wave is exactly 180 degees out of phase with the fast wave?
Cicula to Linea Polaization If we add ight-cicula and left-cicula light, we get E = E 0 B = E 0 c This obviously educes to x ˆ cos( kz ωt)+ y ˆ sin( kz ωt) + x ˆ cos( kz ωt) y ˆ sin( kz ωt) y ˆ cos( kz ωt) x ˆ sin( kz ωt) + y ˆ cos( kz ωt) + x ˆ sin( kz ωt) E = E 0 2x ˆ cos( kz ωt) + 0 B = E 0 c 2 y ˆ cos ( kz ωt ) + 0 [ ] [ ] which is just linealy polaized in the x diection. If we subtact left-cicula fom ight-cicula, we get E = E 0 0 + 2y ˆ sin( kz ωt) B = E 0 c 0 2 x ˆ sin ( kz ωt ) [ ] [ ] which is linealy polaized in the y diection (and out of phase with the oiginal, but nomally you don t see the phase of light)
Cicula Dichoism (Optical Activity) Thee ae some mateials whee the speed of light is diffeent fo left and ight hand cicula polaizations. This is called optical activity o cicula dichoism. These mateials will otate the polaization of linealy polaized light. Fo this to happen, the mateial has to have a helical stuctue of its own that is left o ight handed. Many molecules come in left-handed and ight-handed foms. It is not necessay fo the mateial to be cystalline, even a liquid can show the effect. Nomal chemisty will poduce both foms in equal amounts and the effect will cancel. But biological pocesses based on enzymes often make only one fom o the othe, so optical activity is common. Sugas ae a good example.
Faaday Effect and Cicula Polaization The Faaday Effect is due to the fact that magnetic fields can cause the speed of light to be diffeent fo left-cicula and ight-cicula polaized light in some mateials. The electons in the mateial want to obit the field lines in one diection and not the othe due to the v B Loentz foce. One cicula polaization makes the electons go the ight way, the othe makes them go the wong way, so the dag the electons exet on the light is diffeent fo the two cicula polaizations. We can wite the oiginal linea polaization as an equal mix of left and ight cicula polaizations. The left and ight polaizations get out of phase as they tavel though the mateial. When we add the two cicula polaizations up again, the elative phase shift causes the sum to be linealy polaized light with a diffeent polaization diection. The highe the magnetic field, the lage the velocity diffeence, so the lage the polaization otation.
Cicula Polaization and Complex Exponentials With cicula polaization, we ae constantly dealing with x and y diections, and with phases. Complex exponentials can make the mathematical book-keeping easie. Fist ecall that e iθ = cosθ +isinθ i 2 = 1 Now conside the following expession ( x ˆ ±iˆ y )e iθ = ( x ˆ ±iˆ y )( cosθ +isinθ) = [ x ˆ cosθ m y ˆ sinθ]+i x ˆ sinθ ± y ˆ cosθ [ ] The eal pat of this looks simila to cicula polaization. (the imaginay pat is to, with a 90-degee phase shift) Just like cos( kz ωt) is the eal pat of exp[ i( kz ωt) ], left and ight-cicula polaizations ae the eal pat of [ ] ( x ˆ ± iˆ y )exp i( kz ωt)
Cicula Polaization and Complex Exponentials (2) If we add a phase φ to ight-cicula light, we get [ ( )] = ˆ ( x ˆ iˆ y )exp i kz ωt + φ ( x iˆ y )exp[ i( kz ωt) ]e iφ Add the opposite phase-shift to left-cicula light ( x ˆ + iˆ y )exp[ i( kz ωt) ]e iφ then add these togethe we get [ x ˆ ( e iφ + e iφ ) iˆ y ( e iφ e iφ )]exp i( kz ωt) = [ 2x ˆ cosφ + 2y ˆ sin φ]exp[ i( kz ωt) ] [ ] Now if we take the eal pat of this, we get [ 2x ˆ cosφ + 2y ˆ sinφ]cos kz ωt ( ) So if we add left and ight cicula polaizations with a phase shifts of +φ and φ, we otate the linea polaization by angle 2φ.