Lab 8: L-6, Polarization Lab Worksheet Name This sheet is the lab document your TA will use to score your lab. It is to be turned in at the end of lab. To receive full credit you must use complete sentences and explain your reasoning clearly. Last week in lab you looked at the properties of time-dependent magnetic fields, and how they produced electric fields. In class we said that time-dependent electric fields can generate magnet fields, and that these two coexist in an electromagnetic wave where time-dependent electric and magnetic fields are both present. In this lab you measure an important aspect of an electromagnetic wave: its polarization. Part A gives you some information about electromagnetic waves to help you to understand polarization. A. Electromagnetic waves and fields Suppose an electromagnetic wave is propagating through your area at the speed of light. If you could take a snapshot, you could freeze the electric and magnetic fields at a particular instant in time so that you could look at them. At each point in space around you, and at each instant in time, you would see electric and magnetic fields with these properties: i) r E and r B are always perpendicular ii) E r and B r are always both perpendicular to the wave propagation direction such that E r " B r is in the propagation direction iii) The magnitudes E r " E and B r " B are related as B( x o,y o,z o,t o ) ( 1/c)E( x o,y o,z o,t o ). A1. Consider an electromagnetic wave propagating in the z ˆ direction in a vacuum. The electric field vector of this particular wave is known to point along the y-axis ( E x E z 0), but E y oscillates both in space and time. At some spatial location and time ( x o, y o,z o,t o ) the E-field has magnitude 10 V/m and points in the y ˆ direction. i) What is the direction of the magnetic field vector here? ii) What is the magnitude of the magnetic field vector here?
A2. Consider again the electromagnetic wave of A1, which propagates in the ˆ z -direction. ( ) E y ˆ Suppose that the value r E x o,y o,z o,t o y with E y 10V /m is amplitude of the electric field oscillation, and the wavelength of the wave is 500 nm. What is the E-field at these other locations or times? i) ( x o, y o,z o +125nm,t o ) E x E y E z ii) ( x o, y o +125nm,z o,t o ) E x E y E z iii) ( x o, y o,z o,t o + 0.417 femto " seconds) (1 femto-second 10-15 seconds) E x E y E z The electric field of the electromagnetic wave above can be written E r E o cos( kx "#t) y ˆ with k 2" /# where " is the wavelength, and " 2# / f where f is the frequency. The electric field varies in both space and time. In fact, all points on the wave are moving with the speed of propagation. 2
B. Making and detecting polarized electromagnetic waves In the example of part A, the electric field varied in magnitude and direction, but it always pointed along y ˆ or " y ˆ. This is an example of a linearly polarized electromagnetic wave. Its polarization is usually described with a plane of polarization, in this case the yz plane. The electric field for this wave as it propagates is always in this plane. Since the associated magnetic field is always perpendicular to this, we can describe the polarization completely by describing the orientation of the electric field. Light doesn t often look like this unless it is specially prepared. It is usually a superposition of all possible polarizations, which we call unpolarized light. A linear polarizer can be used to investigate the polarization state. It completely transmits linearly polarized light aligned with its transmission axis, and completely absorbs light linearly polarized perpendicular to its transmission axis. 1. Polarization by reflection. Light is partially polarized whenever it is reflected. You can check this by using the circular polarizer from the polarization kit on the lab table. The transmission axis is marked by raised lines on the black plastic disc that holds the polarizing material. i) Turn on the black gooseneck lab table lamp and look at it directly through the polarizer. Is this light polarized? How did you tell? ii) Now aim the gooseneck lamp at about a 30-45 angle (from the horizontal) at the lab bench, and look through the polarizer at the reflection of the bulb. Is this light polarized? In what direction is the plane of polarization? iii) Polaroid sunglasses operate on the same principle as your kit polarizer. In what direction is the transmission axis of Polaroid sunglasses? Explain. 3
The polarimeter Your polarimeter has two polarizers: The bottom one sits directly over the (unpolarized) incandescent bulb, and is fixed in place. In a two-polarizer system, it is usually called the polarizer, since it produces linear polarized light from an unpolarized source. The top one is rotatable, and usually in-line with the bottom polarizer, but can also be swung to the side. In a two-polarizer system, it is usually called the analyzer, since it analyzes whether a sample placed between the polarizers has changed the polarization of the polarized light from bottom polarizer. 1. Turn on your polarimeter and look down into the top polarizer. Describe the intensity variations of the transmitted light while rotating the top polarizer through 360. 2. Find the transmission axis of the bottom polarizer with the small polarizer (mounted in a black plastic ring) from your box. This one has its transmission axis marked with two ridges in the plastic. Explain your procedure and use a piece of masking tape to mark it on the rim of the polarimeter. Find the transmission axis of the top polarizer in a similar way and mark it also. 3.What is the angle between the transmission axes of the two polarizers when there is: Maximum transmission? What about no transmission? 4
C. The Amplitude of the Transmitted E-Field: θ The light coming from the bottom polarizer is linearly polarized. Consider the case in which this linearly polarized EM wave s E-field vector makes an angle θ with the transmission axis of the analyzer. The analyzer transmits electric fields in one direction (along the transmission axis) and absorbs those that are perpendicular to that direction. For angles not 0º or 90º with respect to the transmission axis the polarizer transmits the component of the E-field along the transmission axis and absorbs the rest. y x 1. Write the space- and time-dependent E-field shown above as a vector sum of two electric field vectors, one oscillating in a plane parallel to the analyzer transmission axis y ˆ, and the other oscillating in a plane perpendicular to the analyzer transmission axis x ˆ. 2. Write down the space- and time-dependent E-field of the EM wave transmitted by the analyzer. What is its polarization angle with respect to the analyzer s transmission axis? Explain. 3. What is the ratio of the amplitude of the E-field of the EM wave transmitted by the analyzer to the amplitude of the E-field of the incident EM wave? Explain. 4. What is the ratio of the intensity after the analyzer to the intensity before the analyzer? 5
5. Put your pencils down, and set the analyzer on your polarimeter so that no light is transmitted. Now insert the small round polarizer from your kit between them and rotate it. Why does light now get through at some angles? 6. Approximately how many degrees of rotation of the small polarizer does it take to go from light to dark? Explain this effect by thinking about the E-field components transmitted at the second and then third polarizer. 6
D. Circular Polarization: Note that above you separated a polarization state into two perpendicular components, one absorbed and one transmitted. This is the basically the same as x and y components of a vector. But there is additional aspect because the x and y components have a time dependence. They oscillate in time sinusoidally. In part C, you added together two components that were in-phase with each other. They were both zero at the same time, and they both peaked at the same time. An important type of polarization occurs when the horizontal component is time-delayed, or phase shifted by one quarter of a period (a 90 phase shift). That is, the horizontal component starts later in time as illustrated by the 3D diagram below. Let both amplitudes be E o here. t o t1 t2 t3 t4 t5 t 1. Use the axes below to draw the components of each polarization as it varies in time. Also draw the vector sum on each axis. y x t o t 1 t 2 t 3 t 4 t 5 2. Describe the time-dependence of the total E-field (direction and magnitude). Does the magnitude oscillate? Does the direction change? 7
3. Now consider the case where the horizontal component is time-advanced, or phase shifted by one quarter of a period. That is, the horizontal component starts earlier in time as illustrated by the 3-D diagram below. Let both amplitudes be E o here. t o t1 t2 t3 t4 t t 5 4. Use the axes below to draw the components of each polarization as it varies in time and draw the vector sum on each axis. y x t o t 1 t 2 t 3 t 4 t 5 5. How does the time dependence of this E-field compare to that of the previous example? 8
The two polarization states above are called right and left circular. They can be described by an E-field that rotates clockwise or counterclockwise with a constant magnitude. 6. Now superimpose these two right and left circular polarizations. That is, add them graphically as you did the two linear polarizations. Suppose they both start out at t o with their E-fields pointing along the y-axis (i.e. they are in phase with each other). y x t o t 1 t 2 t 3 t 4 t 5 7. Write down the space- and time-dependent E-field of the EM wave that is the superposition of these two circular polarizations. What is its polarization angle with respect to y-axis? 9
E. Chiral molecules - sugar: Here you use your polarimeter to look at linearly polarized light propagating through Karo syrup. If you don t know what Karo syrup is, call your mother and ask her. She probably hasn t heard from you in a while, and would enjoy talking with you. We use Karo syrup because it is a very concentrated chiral solution of right-handed molecules (sugars) only. This handedness causes the plane of polarization of linearly polarized light to rotate as it passes through the syrup. The more syrup, the more rotation. Later in the lab, you will explain this by decomposing the linear polarization into right and lefthanded polarizations as in the previous section. Get a set of four jars of Karo syrup (three small and one large, with different heights of syrup). Be careful not to tip them as the syrup will cling to the side and take time to run back down into the jar. 1. Place the large jar (lid off, open end up) on the polarizer and spend some time rotating the analyzer while looking through the polarizers and syrup. You should see colors in the syrup. In what order do the colors appear as you rotate the analyzer clockwise? 2. Why do different colors appear at different rotation angles of the top polarizer? TRANSMISSION ( % ) 3. Place a filter (colored transparency) under the syrup so you are only looking at one color (see spectral transmission below). 100 80 60 40 20 Roscolux #122, Green Diffusion TRANSMISSION ( % ) 0 0 400 500 600 700 400 500 600 700 WAVELENGTH ( nm ) WAVELENGTH ( nm ) Rotate the analyzer so that the light outside the jar but through the filter is extinguished. Now rotate the analyzer until the light through the syrup is extinguished. This is the angle by which the Karo has rotated the linearly polarized light. What direction (clockwise or counter clockwise, looking from the top) is the plane of polarization rotated? How did you tell? (Hint: a little bit of Karo rotates the polarization a little bit, and a lot rotates it more). 100 80 60 40 20 Roscolux #124, Red Cyc Silk 10
Record the polarization rotation for the red and green filters. Filter color: Relative rotation (deg): 4. Do this for the other jars and record all your data below. Sugar depth Green angle Red angle 4. Plot the data for the two colors on the axes below. Use a reasonable scale on the axes and put each color on the graph. Polarization Rotation Sugar Depth 11
5. In the previous section you found that linearly polarized light is a superposition of equal amplitude right and left circularly polarized components in-phase with each other. Below are two circular polarizations with the CCW component delayed by π/2 (90º). They are drawn at various times like in previous examples. What is their sum or superposition state? Is it linear? If so, what is the angle does it makes with the vertical? y x t o t 1 t 2 t 3 t 4 t 5 6. Write down a relation between the phase difference of the left- and right-handed components (" left # " right ), and the angle of the resulting linear polarization with respect to the y-axis (see above). To understand the linearly polarized light as it passes through the Karo, think of it a superposition of right and left-circularly polarized light. As you showed in parts 5 and 6, the polarization angle depends on the phase difference of the circularly polarized waves. Right circularly polarized and left circularly polarized light propagate at different speeds in a chiral solution because they interact differently with the right-handed sugars. Since they travel at different speeds, they arrive at the top of the Karo at different times. This results in a time delay of one component relative to the other after passing through the solution, and a corresponding phase shift. 12
7. Suppose that the right-handed component moves at 0.999999 the speed of the left-handed component (take v left 3x10 8 m/s). Then the left-handed component arrives at the top of the syrup a short time "t before the left-handed component. What is "t after traversing 10 cm of Karo? 8. This time difference corresponds to a phase difference between the right- and left-handed components, determined by what fraction it is of the oscillation period T. Write a relation between the phase shift φ, the time difference Δt, and the frequency of the wave f 1/T. 9. The peak wavelength of red light you used above is 670 nm. What is the phase shift for the time delay calculated in 7? 10. Using the result from 6, what polarization rotation does this correspond to? 11. Combining the ideas of 8-10, write the polarization rotation θ in terms of the wavelength and time delay. 12. Explain whether this describes the sequence of colors you observed in F1, assuming that the time delay does not depend on wavelength. 13. From your data for red and green polarization rotation, determine the actual speed difference for right- and left-handed light in the Karo syrup. 13