Step 1: Introduction: Circular Magnetic Birefringence
• In 1845, Michael Faraday used a powerful electromagnet to rotate the polarization of a beam of light, unifying electromagnetic force with light in one of the most elegant and seminal experiments ever.
• Two decades later, James Maxwell confirmed this with his classical electromagnetic equations which showed that light is actually an electromagnetic wave. It wasn't until the 1960s, however, that Faraday rotation was modeled quantum-mechanically.
• Today, this effect is called the Faraday effect or Faraday rotation. See the last step in this instructable for real-life applications.
About this project:
• A Faraday rotator is a device that rotates the plane of polarization of linearly polarized electromagnetic waves (such as visible light) passing through matter (a "dielectric" - an insulator) by applying a magnetic field parallel to the direction of propagation of the light. (This is a very informative link: http://www.teachspin.com/instruments/faraday/index.shtml )
• I am currently experimenting with the fully functional device that I have built. I made this device to demonstrate Faraday rotation and I am still working on learning the physics behind it in greater depth. This is my senior project for my "WISE" English class. Nice, eh? I get to do Physics for an English class!
• It will cost a bare minimum of $150-200. Take a look at the materials section (Step 3).
• It requires a rather advanced understanding of some topics in physics. The basics are up to you to learn (see bottom of this page to see what they are), but I do my best to explain Faraday Rotation with classical physics in the last steps of this instructable.
For more background Information:
• Watch 20:20 to 21:15 of this video: http://fora.tv/2008/05/01/George_Johnson-The_Ten_Most_Beautiful_Experiments
• This book has a detailed description of the original experiment: The Ten Most Beautiful Experiments (by George Johnson ) (read chapter 6, "Something Deeply Hidden") You can get some of it here: http://books.google.com/books?id=3w83rbjVWZgC&pg=PA75&dq=the+10+most+beautiful+experiments+by+george+johnson+faraday&source=gbs_toc_r&cad=4#v=onepage&q&f=false
• This website has a ton of useful physics that will really help you understand how Faraday rotation works: http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
• Ask the Van is a great reliable resource for answering your physics questions. It is a Q+A that is run by people who really know what they are talking about: http://van.physics.illinois.edu/qa/newquestion.php This is where I first went to make sure that this project would actually be possible to do. You can see my original questions here: http://van.physics.illinois.edu/qa/listing.php?id=17167
• Being familiar with the following terms would be quite useful if you want to understand Faraday rotation: electromagnetic waves, linear polarization, circular polarization, index of refraction, birefringence, lasers, wavelength, dispersion, Verdet constant, and magnetism.
Step 2: Safety FIRST
• I highly suggest you read this before using any laser ever: http://www.repairfaq.org/sam/lasersaf.htm
• Never look into the beam of any laser (this one should be a no-brainer)
• Never point any laser at any person, vehicle, animal, etc. (also a no-brainer)
• Use extreme precaution when using high powered lasers (anything above 5mW)
• Use extreme precaution when using IR (infrared) or UV (ultraviolet) lasers (these wavelengths are not visible to the human eye, so the usual blink reflex cannot protect you if a 5mW beam bounces into your eye)
• Always use proper eye protection -- rated for the correct wavelength and power output of your laser(s)
• Always use a fixed, non-reflective object to stop the beam of any laser over 5mW
• Use a special room with restricted access, no reflective surfaces, and all openings fully covered by thick black fabric for high power
• Read this: http://www.kjmagnetics.com/safety.asp
• Pinch/smash/squish/shatter/severe injury hazard; be very careful. I have a friend who works at a company that puts together powerful magnetic assemblies who nearly lost a hand to a pair of 2" cubes.
• If you want to separate magnets or re-attach magnets, slide them apart. You can put the stack of magnets on the edge of a table and push down to do slide them apart. Never try to re-attach magnets to each other by moving the faces together; this is more dangerous.
• Keep them away from TVs, computers, credit cards, and electronics or they will ruin your stuff.
High Voltage for HeNe Lasers and other things:
• Read this: http://www.repairfaq.org/sam/safety.htm
• Use caution and common sense.
• Turn on/off with the switch on a power-strip.
• For $20, you can make a high voltage power supply that puts out 20-40kV at very low amperage
• Do NOT use neon sign transformers (NSTs) or microwave oven transformers (MOTs). They put out only ~3kV at SEVERAL AMPS. (THIS WILL KILL YOU!)
• Keep one hand in a pants pocket at all times when using any high voltage; this will help limit the amount of current that can go through your heart if you are accidentally shocked.
• Wear proper eye protection
• Never play with the power mains. (DARWIN AWARD MATERIAL )
Machine Shop and Hand Tools:
• Always wear proper eye protection.
• It should go without saying, but use common sense.
• When using power tools (such as drill-press, lathe, mill, grinder, band-saw, table-saw, etc), do NOT wear gloves. The machine can pull in a loose glove AND your hand/arm. If you catch just your flesh in a machine, it will tear cleanly instead of sucking you into the machine. While you operate machines, wear close-fitting clothing, tie back long hair, and remove your jewelry.
• Use proper gloves when you handle sharp metal items and when using hand tools such as a razor-knife.
• Always use set safety guards on machines to the proper height.
• Never work alone. Never work when you are tired or very stressed.
• Work in a very well ventilated room or outside if you are using toxic or volatile paints, solvents, glues, etc.
• Use a paper-filter-type mask for sawdust and other particulates that float around in the air.
Step 3: Materials
This crucial element of the device must be Terbium doped borosilicate glass or Terbium Gallium Garnet (TGG) -- or some similar paramagnetic material with a very high Verdet constant -- in order to produce a measurable effect. The cheapest source for this is http://www.teachspin.com/instruments/US-CA%20prices.shtml#faraday ($100).
1 large, powerful horseshoe magnet:
This can be easily made with the steel from a large electrical transformer and a bunch of really strong neodymium magnets, preferably grade N52. If you use the E-shaped piece of steel from a transformer, you need to cut the middle prong out with a hacksaw in order to make the steel "C" that is the back part of the horseshoe magnet. I got my magnets from a friend who works for a magnet company that tosses a lot of magnets due to small chips or changes in customer specifications. I have also purchased incredibly strong magnets from http://www.kjmagnetics.com/
1 powerful solenoid, 10cm long, with a high current power supply:
As you can see in my photos, I hand-wound a large solenoid on a cardboard tube that I made from a manila folder. The magnet wire came out of the transformer that I cut apart to get the steel for the horseshoe magnet. The solenoid can run on 14.5V DC @ 3A for about 2 minutes before I can start to smell something melting. It certainly would be a more elegant way to make a Faraday rotator -- because you would be linking electricity and magnetism in addition to linking magnetism and light.
1 long, axially magnetized ring magnet:
I have not tried this, but I think that it would be a great way to make a really strong field. However, you have no way to adjust the field strength. You can purchase a stack of 16 of these for $63.25. Based on the helpful field visualizations on the product page (http://www.kjmagnetics.com/proddetail.asp?prod=RC44-N52 ), this would have an average magnetic field parallel to the propagation of the light of an incredible .5 Tesla or so.
2 linear polarizers:
Polaroid sheet works great. I used 4 polarizers because they aren't extremely efficient at polarizing blue light. They are pretty cheap.(http://www.amazon.com/Square-Linear-Polarized-Filter-Window/dp/B004803FAK )
It needs to be able to measures to half a degree.
Lasers: The Faraday effect has different degrees of magnitude for different wavelengths of light.
I bought http://cgi.ebay.com/473-nm-Blue-Laser-DPSS-B-W-Tek-BWB-10-OEM-/220660063507pt=LH_DefaultDomain_0&hash=item33605d5d13 to allow me to experiment with another wavelength of light. It is a beautiful DPSS (diode-pumped-solid-state) 473 nm (turquiose) 5 mW (safe, low power) lab laser. Another reason I bought this laser was to make this project as cool as possible -- as I will be donating the device to the science department at my high school -- as a tool to get students interested in physics.
A 405nm blue-violet laser pointer is a much cheaper source of a shorter wavelength of visible light. However, I found that it doesn't work for this experiment beacuse 405 nm light makes the glass rod fluoresce so intensely yellow that none of the beam actually gets through the rod.
I also bought a 2 mW HeNe (Helium-Neon) laser tube for about $20 on eBay. I hooked it up to my high voltage supply that I made from this (http://www.powerlabs.org/flybackdriver.htm ) or this (http://www.capturedlightning.org/hot-streamer/kv2/kv2/hv/t_fbt/ ) and it worked
well. It puts out 632.8 nm light, which is red-orange.
I borrowed a 532 nm (green) 5mW laser pointer.
I used a 5mW 650 nm (red) laser pointer.
Voltmeter (multimeter) and photo-sensor. I used the power-regulating photodiode from a "B&W Tek BWB-10-OEM" 473 nm laser, but
I'm pretty sure you can find a cheaper replacement. An LED, solar cell, or photocell should work just fine.
• One 2' x 4' piece of pegboard (the sheet with all the 1" evenly spaced 1/4" holes in it) -- this is the standard size.
• Lots of 1" thick pine wood (for mounting polarizers, glass rod, photo-sensor, magnet) Also, if you get a B&W Tek BWB-10-OEM 473 nm
5 mW laser, then you can use this wood (or 3/4"-thick wood) to make a box for the power supply -- you may also want plexiglass or
preferably polycarbonate for the top, woodscrews to hold it together, and hot-glue to put on the brittle plexiglass. If you use wood
screws to put together a box, definately drill pilot holes the size of the screw shank to avoid splitting the wood.
• About 25 regular steel 1/4"-20 nuts.
• About 60 regular steel 1/4" ID washers.
• About 25 regular 1/4"-20 bolts, 1.5" long.
• 2-4 regular 1/4"-20 bolts, ~3" long.
• 2 1/4"-20 wing-nuts.
• 2 1/4"-20 wing-bolts or bolts that you can tighten by hand, 1.5" long.
• Small diameter aluminum pipe for spacers for the 473nm lab laser.
• About 8 small square 2 x 4s and about 16 nails (supports for the main board).
Step 4: Tools
• Hand drill for making holes in the wooden mounts and the pegboard. Only a 1/4"-clearance bit is needed.
• Band-saw for cutting wood, polycarbonate, pegboard, etc.
• Hammer for nailing the wood support blocks onto the pegboard and the steel part of the horseshoe magnet onto the wood mount.
• Hacksaw for cutting wood, pegboard, polycarbonate, nails that are too long, etc.
• Two 7/16" wrenches.
• Pipe cutter for making aluminum tube spacers to mount the blue lab laser.
• Wire strippers for stripping the insulated leads on the sensor.
• Nonmagnetic metric tape or ruler for measuring the distance between the horseshoe magnet and the laser beam in the glass rod.
• Standard/English tape measure.
• Sharpie for marking things.
• Multimeter for measuring the voltage put out by the regulator-sensor diode. I got mine from circuit specialists for $8 (no photo).
• Gauss meter (Hall-effect probe) -- you should borrow this if possible to measure the strength of the magnetic field (no photo).
Step 5: Design and Manufacture of Parts
• A mount for the first linear polarizer.
• A mount for the second linear polarizer and a protractor, made in such a way that the polarizer can move on a hinge that is at the middle of the protractor (the point that would be the center of the circle made by putting two identical protractors next to each other).
• A mount for the glass rod. This can be done any number of ways. I drilled horizontal holes in my mount that now hold 4 zip-ties that hold the glass rod securely in place.
• A mount for the sensor-diode. Just bolt the sensor on if you have the kind I have. Alternatively, you could just use an LED -- it will act as a tiny solar cell.
• A mount for the magnet steel "C." My steel piece from the transformer came with holes in the corners, so I just nailed it on. It is important that this mount can slide on slots in the pegboard so that you can vary the magnetic field and view the Faraday effect.
• A mount for the laser. This may be wood, but for my 473 nm 5 mW lab laser I used aluminum tubing to make spacers -- and I just bolted it right on.
• A poor-man's optical table out of pegboard and wooden block supports that are nailed on the bottom. Make sure it is just the size you want it; too big is better than too small. Hacksaw the pegboard and nail on wooden blocks to the corners with 2 nails each. I added an extra block later near the glass rod mount to stop the board from flexing, but wait until you are done with assembly so that you will not put a nail or a wooden block support where it will get in the way of a bolt.
• A wooden box for the laser power supply. This entails connecting the two "interlock" wires as well as electrical-taping exposed metal and bare wires that carry 120V AC. Polycarbonate is best for making a clear top for the box. Use wood screws to attach the sides of the box; remember to drill pilot holes the size of the screw shank to prevent the wood from splitting.
2) Attaching components to their mounts:
• Hacksaw a slot in the mount for the first linear polarizer; hot glue it in.
• Hacksaw a slot in the mount for the second linear polarizer; hot glue in the protractor and push-tack on the second linear polarizer; make sure that everything is lined up well. Clamp/hold until glue dries/hardens.
• Wait to zip-tie on or otherwise attach the glass rod; it is expensive and easily damaged, so wait until the entire device is assembled.
• Bolt on the sensor to the wooden mount. If you are using an LED, you can drill a hole through the mount and hot glue the LED in the back of the hole (from the back of the mount).
• Put the laser bolts through the laser and through the spacers. It is necessary to use a tin-snips to modify a washer to fit that close to the laser module body.
• Cut aluminum tubing spacers for bolts that will friction-hold the laser power supply box onto the pegboard.
Step 6: Assembly
• Attach bolts with spacers and washers around the box on the pegboard, making sure that it makes a tight friction-fit.
• Bolt the laser module onto the pegboard with long bolts, spacers, and washers.
2) Attach the first linear polarizer.
• Bolt it onto the pegboard in front of the laser module.
3) Attach the glass rod mount.
• Bolt it onto the pegboard in front of the the first linear polarizer; know that the glass rod must line up with the laser beam.
4) Attach the second linear polarizer and protractor mount.
• Bolt it onto the pegboard on one end of the glass rod mount.
5) Attach the optical sensor and beam-stop mount.
• Bolt it onto the pegboard just in front of the protractor.
6) Determine where the magnet will go, cut slots for it to slide on, and attach it.
• Turn on the laser to line things up approximately -- so that the laser beam would go through the glass rod if you were to attach it.
• Line the magnet up with where the glass rod would be so that the poles are equal distances from the ends. Ideally, the poles should be centered a couple centimeters past the ends of the rod. My horseshoe magnet was too narrow (the poles were too close together) and this caused some of the magnetic field to cancel out some of the Faraday effect by rotating the polarization of the light in the ends of the rod in the opposite direction.
• Mark two slots for the bolts to slide on - and cut them with a hacksaw blade (or, if you can get your hands on one, a jigsaw would be preferable). It is best to have the slots go through the holes that are already in the pegboard -- to preserve its structural integrity, so you may need to slot the holes on the glass rod mount and move it to center it with respect to the magnet.
• Bolt on the magnet mount with hand-tightening bolts, in front of the glass rod mount. Always tighten the bolts on the magnet to avoid damage to the glass rod.
7) Attach the glass rod to its mount.
• Zip-tie it on or otherwise attach it -- very securely.
• Never tighten one bolt on a mount all the the way before tightening the other at all. Instead, tighten them each bit by bit, simultaneously.
• Keep things loosened until you are sure that they need to be exactly where they are.
• Don't attach the expensive glass rod until the very end; this will really help protect it from being broken.
• Be very careful when adjusting the magnets on the sliding slots; if you leave the magnet loose it may smash into the glass rod.
• Use wooden-block supports nailed to the bottom of the peg-board to help the optical system maintain its rigidity.
• You may have to slot some things to make them line up just right.
• The holes in the peg-board are exactly 1/4", not 1/4"-clearance. You may want to drill them out so that bolts fit through easily. I did not do this because there are so many holes.
• Use washers on top of wooden mounts and on the bottom of the pegboard.
• Join a FIRST Robotics Competition (FRC) team; this will give you all the skills you will need to make something with your hands and the necessary tools.
Step 7: Aligning and Testing the Optical System
If you have a smoke machine or can borrow one, that could really make things a bit easier, as it will allow you to see the beam of a low powered visible-light laser. Whether you have smoke/fog or not, here is the procedure:
1) Turn on the laser.
2) Line up the laser beam/dot with the optical sensor by adjusting both the position of the sensor mount and the position of the laser, one at a time. Before you tighten the bolts down, make sure that you can see the full width of the beam on the paper beam-stop. While you do this, all other components should be out of the way of the beam.
3) Line up the glass rod so that the beam both enters and exits the rod exactly at the center of the circular end. It may be necessary to slot the mount. Be extremely careful with the glass rod. Tighten down the bolts on the glass rod mount.
4) Make sure that both linear polarizers are positioned so that the beam goes through them. Check that the second linear polarizer can move from about -10 degrees to +30 degrees with the laser beam still going through the actual polaroid part of the square slide -- and not getting blocked by the paper/cardboard part of the square slide.
5) Turn the laser off. You're done.
Step 8: Testing and Troubleshooting
Step 9: Finished! (Now the Data and Calculations.)
• Beta is the angle of rotation.
• V is the Verdet "constant," which tells you about the magneto-optical properties of a specific material at a specific wavelength
• B is the strength of the magnetic field parallel to the propagation of the light
• d is the distance that the light propagates through the material.
Now that I have finished building the device, I am starting to experimenting with it; testing various variables such as:
• wavelength of light -- use different lasers
• strength of the magnetic field parallel to the propagation of the light -- move the horseshoe magnet
• temperature of the rotator material -- I'm not sure how I will do this yet. Extreme temperatures might crack the glass rod.
Measuring the magnet:
In order to vary the magnetic field strength, I am testing the magnets with a Hall-effect probe borrowed at Cornell, nearby. I found that the surface field of my horseshoe magnet is a whopping .57 Tesla. Note that this is the magnetic field strength perpendicular to both the pole-surfaces and the propagation of the light. A Hall-effect probe (Gauss probe) has the property that rotating it will measure the component of the magnetic field in that one direction. (See graph above)
Note that in order to find the average parallel magnetic field at different distances, it is necessary to use the Trapezoid-rule or Simpson's rule to integrate the varying magnetic field over the length of the glass rod, and then divide your total by the entire length. http://mathworld.wolfram.com/SimpsonsRule.html http://mathworld.wolfram.com/TrapezoidalRule.html (See graph above)
I used a Trap-rule, measuring on the axis connecting the poles and finding the average magnetic field; I then moved the Hall-effect probe along the the same axis connecting the poles, stopping when the probe registered a magnetic field equal to the average magnetic field (at the same distance from the axis connecting the poles). Then, I measured every half-centimeter out from that axis (perpendicular to the axis connecting the poles.) Then I graphed the approximate average magnetic field strength as a function of distance from the magnet.(See graph above)
Measuring the Faraday rotation:
Measure the angles of Faraday rotation with the magnet at different distances from the beam, using the same wavelength of light. Plot your angles against the magnet distance. (See graph above)
You can then plot the angles of Faraday rotation against the approximate average magnetic field that you already plotted as a function of distance. The slope should be the Verdet constant of the glass rod material at the particular wavelength, times the length of the glass rod in meters. After editing out 3 points near the edge of the magnetic field approximation, I got 19 radians per Tesla-meter as my calculated value for the Verdet constant of the glass rod for 650 nm light. The known value is 23 radians per Tesla-meter for 650 nm light in this glass rod. My error was only 17.4%. (See graph above)
If you have more than one laser, you can plot the angle of Faraday rotation against the wavelength of the light -- allowing you to plot the Verdet constant of the particular material as a function of wavelength. The Verdet constant is also available from the supplier of the glass rod. I also just found a graph of the Verdet constant for this material (SF-59 glass) as a function of wavelengths. (See graph above)
Learn about how Faraday rotation works (see the next step). Show your Faraday rotator to your friends. Donate it to your high school.
Step 10: Real World Applications
"An optical isolator is a device that allows light to go through in one direction but severely attenuates reflected light propagating in the opposite direction. Modern ultra-high field permanent magnets and special paramagnetic glasses have made these devices quite small, but not cheap (about $2K)."
Optical isolators are applied to optical fiber systems and audio signal switching. They "have important applications in telecommunications preventing reflected signals on fiber optic cables from producing unwanted signals. Isolators are important when lasers are used because reflected light can cause havoc with the operation of the laser itself."
Also, Faraday rotation is used to detect magnetic fields in interstellar space.
To do this, astronomers/physicists have to estimate the electron density of the space that they are examining, and sum up the cumulative effects of all magnetic fields (parallel to the propagation of the light) between the source and where the measurements of polarization are carried out, on Earth.
In interstellar space, the angle of rotation is proportional to the number of electrons that the light hits times the strength of the magnetic field (parallel to the propagation of the light) multiplied by (integrated over) the distance that the light travels in the field, divided by the wavelength of light squared. By comparing and analyzing the relative degrees of polarization at various different wavelengths, astronomers can map out magnetic fields far out in the universe! (At least what they were when the light passed through them, which could have been a very long time ago.)
Step 11: How Things Work: a Classical Explanation
Explanation of what the paper means:
The paper models each electron in the Terbium-glass material as a charged mass on a spring, with a velocity perpendicular to the propagation of the light (and the magnetic field). The spring represents the attractive electric force between the nucleus of the atom and the electron. This ideal spring has an equilibrium position of length equal to zero. (See diagram above)
Light propagates, or moves through space, along the x-axis, by convention. The z and y axes are perpendicular to each other and to the x axis. Light is a combination of an electric field whose magnitude varies sinusoidally with the propagation of the light, and a perpendicular (transverse) magnetic field whose intensity also varies sinusoidally with the propagation of the light. The maximum magnitude of E (the electric field) is equal to the maximum magnitude of B (the magnetic field) divided by c (the speed of light in a vacuum, 299,792,458 m/s). Visible light is one type of electromagnetic wave. (See diagram above)
Linearly polarized light is an electromagnetic wave whose electric field is confined to one plane and whose magnetic field is confined to another plane that is perpendicular to the first. Linear polarized light is thus called a plane wave. (See diagram above)
Circularly polarized light is composed of two plane waves of equal amplitude but differing in phase by 90 or 270 degrees. The sum of the two periodic electric field vectors gives a net electric field vector with constant magnitude, and that is rotating either clockwise or counter clockwise (both from the point of view of the light source), depending on whether the phase difference is 90 degrees or 270 degrees. Right-handed circularly polarized light and left-handed circularly polarized light have electric field vectors that spiral in opposite directions as the light propagates through space. (See diagram above)
When working with the actual device, the linear polarization is considered to be the direction of the Z-axis (the one with the electric field). An equal combination of right and left handed circularly polarized light waves, in phase with each other, add up exactly to a linear polarized light wave. This is because the horizontal plane waves are 180 degrees out of phase with each other and will thus cancel each other out. The vertical plane waves, which are in phase with each other, just give a net vertical plane wave that has twice the amplitude.
The electric field applies an attractive or repulsive force on the electron. Because the rotating field vector of the circularly polarized light is periodic in nature in the zy plane (perpendicular to the propagation), it will apply a periodic force on the electron, causing it to experience a periodic displacement in said plane. This periodic displacement occurs in opposite directions for right hand and left hand polarizations, because the periodic electric field is cycling in opposite directions. This results in the electron being displaced some distance from its equilibrium position at all times.
However, linearly polarized light would only cause a periodic up/down motion of the electron on the “spring.” If one averages the net displacement of the electron over time, it would be zero. Thus, the electron just periodically gains and loses energy, with a net change in its energy of zero.
The amplitude of the electron’s angular motion (or equivalently, in the case of linearly polarized light, the displacement from the electron’s equilibrium position) is directly proportional to its energy; the energy is stored in the “spring.” Thus, when the electric field of the circularly polarized light wave acts on the electron, it must either add or subtract (depending on its handedness) from the electron's energy.
This energy that the electron gains is lost from the light, causing the amplitude of the light to decrease (or the wave dissolves altogether). Normally, with no external magnetic field present, the linearly polarized light wave is re-emitted by the periodically moving electron. The periodically moving electron naturally creates a self-propagating electromagnetic wave: it creates a periodic electric field due to its charge and periodic motion, and a transverse periodic magnetic field due to its charge, velocity, and periodic motion. This causes the "spring" to return to its original position, the electron to again reach its equilibrium position, and the electron to lose its previously gained energy. This light wave is identical to the original right or left handed polarized light wave that first interacted with the electron.
If there is a magnetic field, the re-emitted wave will be different from the absorbed wave. Because the affected electron has been given a velocity perpendicular to the applied magnetic field by the periodic electric field of the circularly polarized light wave, it experiences a force up that stretches the "spring" a little bit farther up or down. (Remember that the electron is already experiencing a circular periodic motion and has an unusually high or low energy due to the periodic electric field from the circularly polarized light; the "spring" has already been stretched by the light.) The force on a moving charged particle is equal to qvB, which is the charge of the particle times the velocity of the particle times the magnitude of the magnetic field that is perpendicular to the velocity of the charged particle. This force is what causes the small increase or decrease in the electron's energy, depending on the direction of the force -- which depends on the direction of the electron's velocity -- which depends on the handedness of the circularly polarized light that interacted with the electron.
When the "spring" pulls the electron back to its original equilibrium position, the electron loses energy and emits an electromagnetic wave, just like in the case discussed earlier where no external magnetic field is present. However, depending on the handedness of the light, the electron falls a slightly different distance (the "spring" releases a slightly different amount of energy). Therefore, the circularly polarized light waves that are released have slightly different wavelengths, as the energy of an electromagnetic wave is inversely proportional to its wavelength.
The index of refraction is n, which equals c, the speed of light in a vacuum, divided by the speed of the light propagating through the medium. As do many materials, the glass rod medium inherently has different indexes of refraction at different wavelengths of light. This can be explained by the different losses of energy from the oscillating electric field to the electron, which is made to oscillate. The closer the frequency of the light (a constant times one over the wavelength) to the natural resonant frequency of the electron as a mass on a "spring" (depends only on mass and spring constant), the more energy lost to the oscillating electron and the higher the amplitude of the electron. The higher the amplitude of the electron's periodic motion, the longer it takes the electron to re-emit the absorbed electromagnetic wave, and thus the longer it takes for the light to propagate through the medium a certain distance. The slower propagation is equivalent to a higher index of refraction. The phenomenon of different indexes of refraction existing for light at different wavelengths in a medium is called dispersion.
So, when one gets the two aforementioned circularly polarized waves with slightly different wavelengths propagating in such a medium, due to the applied external magnetic field of the permanent magnets, one travels slightly faster than the other due to dispersion. Finally, this in turn causes these two waves to become slightly out of phase.
The sum of two circularly polarized waves of equal magnitude that are out of phase is a rotated linearly polarized wave and a leftover left or right handed circularly polarized component. When this mix of waves reaches the Polaroids, only the rotated linearly polarized light can pass through, and the leftover circularly polarized component is screened out. Thus, one can observe the apparent rotation.
Ultimately, it is the spiraling electric field vectors of the two polarized light components of the linearly polarized light that give the electrons their velocities in the zy plane and thus allow the applied external magnetic field to act on the electrons, causing them to lose slightly different amounts of energy and to thus re-emit light at slightly different wavelengths. Then, due to the natural dispersion of the medium, the two circularly polarized waves become out of phase; they then will add to net a slightly rotated observable linearly polarized wave component and a small left-over right or left handed circularly polarized component that cannot pass through the second linear polarizer.