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Waves,Heat&Light Lab

by: Lamont Barton

Waves,Heat&Light Lab PHYSICS 341

Lamont Barton
GPA 3.91


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This 17 page Class Notes was uploaded by Lamont Barton on Thursday October 29, 2015. The Class Notes belongs to PHYSICS 341 at University of Michigan taught by Staff in Fall. Since its upload, it has received 13 views. For similar materials see /class/231645/physics-341-university-of-michigan in Physics 2 at University of Michigan.


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Date Created: 10/29/15
Physics 341 Experiment 4 Page 41 Chapter 4 Polarization 41 Introduction Polarization generally just means orientation It comes from the Greek word polos for the axis of a spinning globe Wave polarization occurs for vector fields For light electromagnetic waves the vectors are the electric and magnetic fields and the light s polarization direction is by convention along the direction of the electric field Generally you should expect fields to have three vector components e g xyz but light waves only have two nonvanishing components the two that are perpendicular to the direction of the wave In this experiment we will be concerned with the polarization of light The most elegant tests of quantum mechanics have been performed by measuring correlations of the polarization of photons from atomic transitions This is beyond the scope of these experiments but it s good to keep in mind that some of nature39s thomier problems are lurking in the corners Electromagnetic waves are the solutions of Maxwell s equations in a vacuum VE0 VB0 BB V Ei 41 x at BE VXB8 7 0H0 at In order to satisfy all four equations the waves must have the E and B fields transverse to the propagation direction Thus if the wave is traveling along the positive zaxis the electric field can be parallel to the xaxis and Bfield parallel to y Half a cycle later E and B are parallel to 7x and 7y Since the fields oscillate back and forth several hundred trillion times per second we don t usually know their sense 139e x vs 7x Polarization of light therefore only refers to direction eg x not sense If the light propagates in the opposite direction along 72 then the E and B fields are instead respectively parallel to y and x The direction the light travels is determined by the direction of the vector cross product EgtltB In the following sections we will try to explain polarization phenomena in terms of both mathematical expressions for wave amplitudes and symmetry When applicable symmetry arguments are the simplest and usually easiest to understand However as situations become more complex it is useful to have a mathematical description robust enough to cover any conceivable physical arrangement The first tool we need is a way of describing plane waves traveling along the zaxis of Cartesian space with wavelength 1 and frequency f Ezt E0 coskz wt 42 Physics 341 Experiment 4 Page 42 where k 275 l and a 279 In this expression cos could be equally well replaced by sin What is important is the relative sign of the z and l arguments If zct wkt as time advances the phase of the wave remains constant This is a plane wave traveling in the positive z direction at velocity 0 Conversely Ezt 2 E0 coskz wt describes a wave traveling in the negative z direction This could equally well be described by Ezt 2 E0 cos kz al The only criterion is that the sign of the z and l terms are the same for this backward propagation direction 4 099 G K 0 419 l gt X E0 O u 6 f E Figure 41 A wave polarized along the xdirection can equally well be represented by the coherent sum of amplitudes along the x39 and y39 axes In what follows we will ignore the magnetic eld B since its value can be immediately inferred from the functional form for E by applying Maxwell s equations Thus it adds no additional degree of freedom to the range of allowable solutions The following experiments include a takehome kit of polarized materials as well as a series of exercises that must be done in class because of the additional equipment requirements The kit should contain the following items 1 Three linear polarizers grayish 2 l2wave plate transparent 3 l4wave plate transparent 4 Rightcircular polarizer grayish 5 Leftcircular polarizer grayish 6 Glass microscope slide plastic box piece of aluminum foil Physics 341 Experiment 4 Page 43 7 Red green and blue plastic lters As you go through the various parts of this experiment take careful notes in your lab notebook and label each part clearly by section number 42 Linear Polarization A beam linearly polarized along the xaxis and traveling in the positive z direction can be represented by Ezt E01 coskz wt 43 where f is the unit vector along the xaxis Of course the choice of coordinate system is completely arbitrary If we have a second coordinate system rotated by an angle 6 about the z direction see Figure 41 we would represent the same beam by E39z t E0 cos9 cosltZ atx E0 sin9 coskz wt 44 Although in this primed reference frame there are both x and y components the wave is still plane polarized because the space and time dependence of the two components are identical This relationship can obviously be inverted 7 if you know the two components of the amplitude in the primed frame you can find the rotation angle of the polarization direction The function of a linear polarizer is to transmit only the amplitude parallel to the axis of the polarizer If this direction is parallel to the xaxis only the x component of the field will survive and the y component will be removed For incident light with random polarization only half will survive This is why linear polarizers always look gray under normal illumination for example Polaroid sunglasses Now suppose we superimpose a second linear polarizer at right angles to the first so light is only transmitted with E parallel to The joint transmission of the two will be zero because the output of the first will be completely attenuated by the second We can relax this condition and ask what happens for intermediate angles between polarizers Suppose that the allowed polarization direction of the second polarizer is set at angle Qto the first In the preferred frame of the second polarizer the E field is given by Equation 44 Only the component parallel to is will be transmitted by this second filter and so the amplitude through the pair will be proportional to cos9 For an ideal polarizer the light intensity is proportional to the square of the amplitude so that the dependence of intensity with polarizer angle is 190 cos29 45 This is called Malus s law 0 Verify Malus s law using an incandescent light source and the two 5 cm diameter linear polarizers Mount one in a 2 lens holder and the other in the rotary mount Use the photodiode and HewlettPackard DVM to measure the light intensity in the same way as Physics 341 Experiment 4 Page 44 performed in Experiment 2 Put a green interference lter in the light beam just in front of the photodiode The polarizers are most ef cient in this part of the spectrum Use a lamp voltage of 10 V and a spacing 30 cm between the light source and the detector Keep the polarizers at least 5 cm from the bulb or they will be ruined For the following use the two linear polarizers from your kit after you have identi ed them as explained in the next section 43 With the two polarizers crossed so that minimum light is transmitted interpose a third linear polarizer at 45C in the middle What happens Explain why Under the same conditions as above insert various transparent objects such as plastic boxes eyeglasses microscope slides with transparent tape etc What do you nd Polarization by Re ection Your polarized light kit has ve grayish lters of which three are linear polarizers and two are circular polarizers In this part of the experiment you should separate the two different kinds of polarizers and mark the transmission axes on the linear polarizers Lamp You need a table with a lamp or a window near the far end as shown in Figure 42 Put a magazine or a book with a shiny cover on the table and put the microscope slide from the kit on top Look at an angle such that the re ection of the light makes it dif cult or impossible to read the print in the magazine andor to read the print below the glass microscope slide Rotate Filter Flip Magazine a b Figure 42 Glare allows determination of polarization transmission axis Now try each of the grayish lters in front of your eye Rotate each lter as shown in Figure 42b to see if you can nd a position of the lter in which it will best reduce the glare You can then move the magazine and your head until the glare reduction with the lter is most dramatic Use that setup from now on When you rotate the lter through 90 the glare should become very bad again Now mp the lter over so that you look through the opposite side Rotate the lter after ipping it over and try to nd out if it Physics 341 Experiment 4 Page 45 makes any difference whether you look through the filter from one side or the other For the linear polarizers ipping should have no effect For circular polarizers the behavior is distinctly different Now you should be able to separate your five filters into a group of three linear polarizers and two circular polarizers List in your notes all the features you found that distinguish the two groups Hold each of the three linear polarizers in front of your eye so that it best reduces the glare and then mark with a vertical line on small circular paper labels that you can attach to each filter Now you should be able to pick up each filter with the line in a vertical position and get the best glare reduction Your line marks the lter transmission axis Now that you are a beginning expert gather a variety of objects of di erent materials put them in place of the magazine and record what you observe with a linear polarizer Examples a piece of paper on which you have written a few words with a soft No 2 pencil a shiny metal surface the plastic box of your kit and other objects Which objects shown polarization effects The ability of polarizing filters to preferentially block re ected light from water and other smooth surfaces makes them valuable for sunglasses they reduce glare far better than the unpolarized variety 44 Polarization by Scattering On a clear day if you look at the blue sky through a polarizer you can turn the polarizer so that the sky looks darker and white clouds stand out beautifully Photographers use polarizing filters for this purpose The light from the sky is partially polarized but you must look in the right direction for a maximum effect the quotmidway beltquot in Figure 43 The blue color of the sky is due to the fact that the short wavelengths blue of the light from the sun are scattered more effectively by the atmosphere than the longer wavelengths red Physics 341 Experiment 4 Page 46 Zenith Midway belt 39 Anti sun Figure 43 Diagram for observing the polarization of sky light In the following experiment let light from an incandescent lamp pass through water with a few drops of milk to scatter the light If you look at the water through a linear polarizer you can see that the light intensity varies as you turn the polarizer You can use this to nd the direction of polarization The incident light is unpolarized but the light scattered out of the liquid is obviously partially polarized The scattered light is produced when electrons in the liquid are caused to oscillate transverse to the beam direction by the E eld of the incident light At a 900 scattering angle these vibrations can only have components perpendicular to the scattering plane or along the scattered direction Since the Ef1eld must vanish along the propagation direction the latter contribution must vanish Verify that the polarization direction is consistent with this explanation The scattered light is only partially polarized because part of the light is scattered more than once In such multiple scattering the light does not remain in one plane The line of sight is no longer always perpendicular to the direction of propagation and multiply scattered light is less polarized 0 Look through a linear polarizer at the milky water through which you shine light Turn the polarizer and nd the direction of polarization Put a linear polarizer between the lamp and the water and rotate it You can thus verify which polarization direction is responsible for the scattered light 45 Optical Activity If you want to do this experiment at home you should mix approximately one part of clear Karo corn syrup and one part of water in a glass or better a square bottle However we also have square bottles with Karo etc available for use in the labs 0 Mount a linear polarizer behind the bottle and another linear polarizer in front as shown in Figure 44 Use the rotary mount for one of these so you can measure angles carefully Let light shine through both filters and through the solution turn the filter nearest you Physics 341 Experiment 4 Page 47 and record what you see The optical rotation depends on wavelength use the color lters in your kit to determine the rotation for red green and blue light 0 Without the solution between the polarizers or with plain water extinction would occur when the polarizers are crossed You will notice that this is no longer true with the corn syrup inserted between The solution rotates the direction of polarization ie it is optically active You should nd out whether you now need to rotate the nearer lter clockwise or counterclockwise to get extinction Record this angle and other experimental results in your notes Note To verify that the rotation is say 20 clockwise rather than 160 CCW you should gure out a way to vary the thickness of solution Fruit sugar fructose or levulose and turpentine rotate in the opposite direction how about honey Note The rotation is due to an asymmetry in the corn syrup molecules which can be right handed or lefthanded Only living organisms produce this handedness this is one of the mysteries of life Almost all amino acids produced by living organisms here on Earth have a lefthanded optical activity Amino acids found in certain meteorites are equal mixtures of right and lefthanded molecules This has been used as evidence of their extraterrestrial origin Thus if you are a health worker and someday take a blood sample from someone and it exhibits righthanded optical activity use caution You may be dealing with an alien Lamp Figure 44 A corn syrup solution rotates the direction of polarization It has been found that the odor associated with chemically indistinguishable molecules is correlated with optical activity For example one molecule whose solution causes a right handed rotation of polarized light has an orange odor Its mirror image structurally causes left handed rotation and has a lemon odor Strange but true Physics 341 Experiment 4 Page 48 Chemists use devices called polarimeters to examine and measure the optical activity of chemical compounds This can be used to identify compounds etc Also since optical activity changes when a magnetic eld is applied the socalled Faraday effect optical and radio astronomers study polarized light and radio waves to deduce the magnetic elds in stars and galaxies 46 Circular Polarization and Re ection Your takehome kit has two grayish circular polarizers You should already have identified and marked the 3 linear polarizers and therefore the two that remain must be circular polarizers if nobody goofed Here is one way to check 0 Put one circular polarizer on top of a shiny not a painted metal surface mirror knife blade scissors a dime Flip it over When you have the correct side up the dime will look very dark Mark the up side with a C Do this with both circular polarizers Prof Jens Zorn of the University of Michigan says that he has won much money betting other physicists at meetings that they will not be able to explain why the metal looks dark through the filter Compare these results with what you observe by doing the same experiment with a single linear polarizer If you put the two circular polarizers back to back with the two sides marked C on the outside of the sandwich very little light will go through no matter how you rotate the filters 7 if you indeed received one L and one R left and right filter as you should have To create circularly polarized light we must be able to somehow control the relative phase of the x and y components Fortunately there are various materials that are slightly anisotropic Because of their atomic structure linearly polarized light will travel at different velocities along two transverse axes By choosing the appropriate thickness one can make a l4wave plate sometimes called a N4plate that introduces a 900 phase shift or a l2wave plate sometimes called a NZplate that introduces a 1800 phase shift Assume we start out with light polarized along a 450 line to the xaxis coskz atx coslltZ aty 46 E 5 After traversing a quarterwave plate aligned with its crystal axes parallel to i and y a net phase E0Zt shift of 900 will occur so that the wave can now be represented by E A E A focoskz 600x focoskz cot 90quot y 5 E E E focoskz am i fosinkz am 2 5 E0 zt 47 Physics 341 Experiment 4 Page 49 Thus we have created circularly polarized light The polarization rotation direction right or left will depend on which way the axes of the quarterwave plate are oriented Note Both optical activity and circular polarization can be left or righthanded but the two phenomena are completely different otherwise A sugar solution rotates the direction of polarization but you still get linearly polarized light out A circular polarizer does not produce linearly polarized light A l4wave plate does not rotate the direction of polarization as optically active materials do Section 45 It is the electric field vector that rotates at the frequency of light in circular polarization 0 Notice that the order in which light passes through the filters is important You must rst linearly polarize the light before applying the 900 phase shift If you do it the other way around you just get linearly polarized light You can check this with your filters What happens to circularly polarized light when it passes through a linear polarizer Try to explain why Evidently from what you have just seen there is a distinct difference between circular polarization and linear polarization Recall that Equation 43 described a wave linearly polarized along the xaxis The equivalent forms for right and left circularly polarized waves are respectively E t E0 cosk wtx E0 sink CODA and z Z i Z 0 5 5 y E E 48 E t iocos k a7t x iosin k at A 0z 15 z 15 z y Although these equations look similar to Equation 44 the behavior is somewhat different In Equation 48 the x and ycomponents are 900 out of phase 7 when the xcomponent is maximum the ycomponent is zero and viceversa For linearly polarized light the x and ycomponents are exactly in phase and therefore rise and fall together The next question is what happens when circularly polarized light is re ected by a metal surface At the surface the response of conduction electrons to any applied electric field guarantees that the sum of electric fields from the incident and re ected rays must always equal zero If an incoming rightcircular ray can be represented by E1zt coskz am sinkz 60057 49 E E the reflected ray must be E E ER 2 t 70cos kz am ism Ia my 410 J5 Physics 341 Experiment 4 Page 410 so that at the metal surface 20 the sum of the two elds E1 and ER is identically zero We would like to characterize the polarization of the re ected beam described by Equation 410 To do this we need a new set of coordinate axes that correspond to light traveling in the positive direction The required transformation is 56 3 5 411 2 i The sign change for 2 gt 239 has the desired effect of describing a wave moving in the positive 239 direction the sign change i a 39 maintains a righthanded coordinate system ie x gtlty z With these transformations the re ected wave is gcoskz39 atx39 sinkz39 aty 412 E 5 Compared to the original wave there is a sign change for the ycomponent of the electric field This corresponds to a reversal from an incident right circular polarization to a re ected left circular polarization This is the reason the back re ected light from a metal surface is completely absorbed by a circular polarizer this trick is sometimes used to reduce glare from external light sources that re ect from the surface of computer display monitors ERZI 2 There is a more graphic way of viewing what39s going on here The motion of circularly polarized light can be likened to the twisting of a screw The process of re ection in a mirror turns a righthanded screw into a lefthanded screw and vice versa 47 Make Your Own Circular Polarizer Your kit has two clear filters and one 25 X 75 cm microscope slide One of the filters is a U4 wave plate and the other is a l2wave plate hopefully As we have discussed if we put a linear polarizer next to a l4wave plate we can get a circular polarizer if we do it the right way From the techniques introduced in Section 46 you should be able to Identify the l4wave plate Make your own R polarizer Make your own L polarizer Note Since you don39t know which is the fast axis on the l4wave plate you can t know whether you make an R or L polarizer Never mind Arbitrarily call one R and one L In Section 49 you will perform a simple experiment to find out which is which Physics 341 Experiment 4 Page 411 0 Test whether you did all these steps correctly Mark the l4wave plate Does it make any difference which side of the l4wave plate is next to the linear polarizer in your homemade circular polarizer Flip it 0 Now that you know how to make your own circular polarizer you may want to check whether the circular polarizers in your kit are made similarly namely by bonding a linear polarizer to a l4wave plate In fact if you followed Section 43 conscientiously you already have the answer Is light that passed through a circular polarizer linearly polarized How would you find out Record you answer in your notebook 48 Determining the Handedness of Light Determining which of your circular polarizers is left and which is right is easy once you know some basic physics about how your l4wave plate works The l4wave plate and l2wave plates in your optics kit are really nothing more than a bit of stretched plastic All plastic is made of long chain molecules known as polymers Plastic gets its plastic properties because these molecules are all tangled up together like cooked spaghetti Figure 45 Cooked spaghetti or unstretched polymer plastic If you stretch a piece of plastic wrap or a plastic bag it deforms it actually changes its shape long before it tears This is the very property that we call plastic deformation in fact At the molecular level the long polymer molecules don t change their length but they do all start to pull into alignment So stretched plastic has the polymers moreorless lined up Figure 46 Stretched polymer plastic or stretched cooked spaghetti Physics 341 Experiment 4 Page 412 Light that is polarized parallel to the lined up molecules sees a higher index of refraction than light polarized perpendicular to the molecules Parallelpolarized light consequently moves slower through the plastic The stretched axis is therefore called the slow axis in the plastic The axis perpendicular to the stretch direction is called the fast axis Physically this is because the electrons in the polymer respond more to the electric elds in light when those elds are directed along the bonds between atoms molecule than when the elds are perpendicular to it In Lab 3 you learned that light travels at different speeds through different materials Now here is something new light with different polarizations travels at different speeds in the same material if the material is anisotropic If light going through the plastic is polarized in between the fast and slow directions then the fast and slow components of the polarization vector each do their own thing The component of electric eld along the slow axis travels more slowly than the perpendicular component traveling along the fast direction The slow component falls behind the fast component In a l4wave plate the slow component falls behind the fast component by a quarter of a cycle That means that if the light entering oscillates like cos0t the slow component will come out oscillating like sin0t This is precisely the required property to turn linearly polarized light Equation 46 into circularly polarized light Equation 47 o The gure below shows linearly polarized light entering a l4wave plate Copy this gure into your notebook and label the following 0 Show the fast axis 0 Show the slow axis 0 Draw an arrow on the circle showing the direction of circulation of the electric eld after it leaves the polarizer The light is incident from the lower left Physics 341 Experiment 4 Page 413 Figure 47 Linearly polarized light passes through plastic stretched at 450 with respect to the polarization of the light On the other side the light is circularly polarized There are two different systems for labeling the circular polarization of light The rst method uses the righthand rule Using your right hand point the thumb in the direction of the light propagation Your ngers will then curl in the direction of motion of the rotating electric eld if the light is righthanded RH If the electric eld rotates the other way the light is called left handed LH This method of labeling right and left polarization is called the handedness convention 0 After the light in Figure 47 passes through the quarterwave plate is it right handed or left handed The other convention for labeling the light uses the similarity between the spiral corkscrew shape of the electric eld vector along the aXis of propagation of the light and the spiral of the threads on a screw This is called the screw convention Light whose electric eld spirals through space like an ordinary screw is called right circularly polarized RCP because the thread spiral on ordinary screws is conventionally called a righthanded thread If the electric eld spirals in space like a left handed thread it s called left circular polarization LCP Now here s a curious fact you can discover on your own 0 Using diagrams of the corkscrew shape of a circularly polarized electric field drawn in your notebook show that righthanded RH light according to the handedness convention is left circularly polarized LCP according to the screw convention Also show that lefthanded LH light is right circularly polarized RCP So two reasonable and intuitive conventions happen to use the opposite words right and left to describe the same state of circular polarization Don t get confused Most physicists use the handedness convention but optical physicists know both and now so do you The screw convention is based on a snapshot of a light wave at any instant the electric eld traces out either a lefthanded or righthanded spiral The handedness convention is based on how the direction of the electric field varies with time as the circularly polarized light passes a particular point in space 49 Construct your own 14Wave plate Cellophane tape NOT Scotch Magic tape is an excellent plastic material for constructing your own l4wave plate 0 Stretch some clear cellophane tape using your fingers and then stick it down to the microscope slide in your optics kit This will take a little practice but you should be able to stretch it just enough to make a region with good l4wave retardation Use a narrow strip so you can stretch it quire a bit Clear polyethylene or food wrap also works but you may have to stretch it until it almost breaks Now combine this with a linear polarizer to make a polarizer for righthanded light Test your 1Awave plate using re ection as described in Sect 47 Physics 341 Experiment 4 Page 414 You are now ready to use your new polarizer to make a real measurement But first take a look at the following list of facts and make sure that they make sense to you based on what you have observed and learned Some Facts about Circular Polarizers l N E 4 V39 To produce circularly polarized light the light must go through the linear polarizer first then the l4wave plate If you send it through the other way you get linear polarized light To analyze circularly polarized light the light must go through the l4wave plate rst then the linear polarizer A righthanded analyzer will not pass lefthand polarized light and viceversa If circularly polarized light is re ected the handedness reverses l4wave plates are true quarter wave retarders only for one wavelength usually chosen to be green light because it is near the center of the visible spectrum For other wavelengths the plate will shift the two polarization components by more or less than 90 Since a circular polarizer contains a l4wave plate it will work best at the design wavelength of the l4wave plate while for other wavelengths the transmitted light will be elliptically polarized Thus if you put an L and an R polarizer backtoback instead of complete extinction you get some blue light transmitted Linear polarizers do not involve l4wave plates and give nearly complete linear polarization over the entire visible spectrum Physics 341 Experiment 4 Page 415 0 Use you homemade polarizer to test the two circular polarizers in your optics kit Now you can tell which is which Label the appropriate polarizers RH or LH Note you could also label them RCP and LCP if you want but remember the screwy convention is opposite to the handedness convention TAPE THE LABELED CIRCULAR POLARIZERS INTO YOUR LAB BOOK SO THAT WE CAN CHECK IF YOU DID THIS RIGHT 410 The 12Wave Plate The 12wave plate has a fast and a slow axis just like the 14wave plate but it does not produce circularly polarized light Let us imagine a linear polarizer at 450 in front of the 12wave plate The component of E along the slow axis is retarded by onehalf cycle or 180 If the light beam after the first polarizer is described by coskz atx coskz aty 413 Then following the 12wave plate the field is E0Zt E0 A E0 A E0zt icOSUCZ COOX ECOSUCZ Coljy 414 5 Try such a combination of linear polarizer and 12wave plate and analyze the direction of polarization of the light that went through it What happens when you rotate one lter Record you result you need two linear polarizers one in front and one behind the 12wave plate Explain What happens to circularly polarized light when it passes through a 12wave plate Verify your prediction with a simple test using the components in your optics kit Explain 411 Re ection Again Now try to predict what will happen when you slip a 14wave plate between the circular polarizer and the metal surface Record your prediction then try Explain the result Can you predict what will happen when you slip the 12wave plate between the circular polarizer and the metal or both the 14 and 12 Again record your prediction then try it Can you predict what will happen when you pass linearly polarized light through your sugar solution then re ect it in a mirror and send it back through the sugar solution Record your prediction and try it Physics 341 Experiment 4 Page 416 Get a piece of aluminum foil and put one of your circular polarizers on top of it so that the foil looks dark blue You are already familiar with this effect Now lift the polarizer slightly off the foil so that you can see the shadow of the polarizer on the metal Why is only the shadow dark Make a Vshaped crease in the aluminum foil Use illuminations such that most of the light comes from a definite direction a lamp or a window Put the circular polarizer back on the creased foil Does the crease look light or dark Explain Hint When you look at your right hand in a single mirror it looks like a left hand What does it look like in a double mirror made by joining two mirrors at right angles If you look through a linear polarizer at light re ected off your plastic box you will see colors Even without the polarizer you can see faint colors if the light is right Explain Physics 341 Experiment 4 Page 417 Appendix 4A Optical Rotation Assume a beam of light linearly polarized along the y axis EL E0 coskz at 415 where k 272 L and a 27y Circularly polarized light can be represented by the expressions Em E0 cosltZ mm sinlltz am 416a Em E0 cosltZ am sinkz am 4 16b Thus the linearly polarized beam given in Equation 415 can be rewritten as 1 1 ELP ZEERCP EELCP 417 Now imagine this beam enters a medium which has slightly different propagation velocities for light with opposite circular polarity After some distance the phase of one wave will be slightly advanced with respect to the other EU coskz at 5 sinkz at 5y E 418 70 cosltZ 6005K sinkz 60037 Since cosacos 2cosaB JcosaB 419 sina sin 2cosaB JsinaB we can rewrite Equation 418 as 6 6 A 5 A EL E0 coskz cotEIcosEx s1n3y 420 This is a plane polarized wave with the polarization twisted by an angle 52 from the x aXis towards the y aXis This effect called optical rotation occurs whenever light passes through a substance containing molecules with a preferential screw direction Such constituents are omnipresent in biological systems and their byproducts


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