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little gems Representing Circular Polarization with a Box of Cereal Eugene Torigoe, Allegheny College, Meadville, PA 16335

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ircular polarization is a phenomenon that is usually represented using complex perspective images. It is a shame that circular polarization is so difficult to understand because it is a very interesting topic that has many applications, including current 3-D movie technologies.1 I have developed a three-dimensional model that can be used to more easily demonstrate how a linearly polarized beam can be transformed into a circularly polarized beam. The model represents two equal magnitude electric field components, which were a result of the polarization vector that is 45° between the fast and slow axes of the birefringent crystal (see Fig. 1). Because each axis in the crystal has a different index of refraction, one component of the electric field will travel faster relative to the other component. If the relative phase shift of the components is p/2 rad (1/4 of a wavelength), then the exiting beam will be circularly polarized (see Fig. 2). Crystals with the correct width to create a p/2 rad relative shift are called “quarter-wave plates.” The model depicts this phenomenon by allowing the vertical component to shift relative to the horizontal component by 1/4 of the wavelength. The model can be made by gluing or printing the model template onto cardboard from a box of cereal, or cardstock paper based on availability. The template as well as directions for construction can be found online.2 I have found that the model can be a useful tool when stu-

Fig. 1. The beam is linearly polarized. Students should visualize the sum of the two waves yield a wave that oscillates along the diagonal. The two pieces shift relative to one another to demonstrate how a circularly polarized wave is formed.

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dents are explaining ideas about polarization to one another. You can ask your students to use the model to understand other related phenomena such as the effect of a half-wave plate, elliptical polarization, and the difference between left and right circular polarization. While the primary purpose of the model is to demonstrate the formation of circularly polarized light, it can also be used in conjunction with other common classroom demonstrations of light phenomena. While commercially available quarter-wave plates are expensive, many varieties of transparent tape exhibit birefringence and can be used to produce circular and elliptically polarized light.3 The colored mosaic created by randomly overlapping pieces of tape between two polarizers depends on the process described by this model.4 For courses that only cover linear polarization, this model may also be used to demonstrate how a wave can be broken up into components. Further, by relabeling one of the components, one can use the model to demonstrate the perpendicular E and B fields of an electromagnetic wave. Although in the case of the electromagnetic wave, the E and B fields must always be in phase. References 1. 2. 3. 4.

Heldrun Schmitzer, Dennis Tierney, and Terry Toepker, “Real 3-D: How does it work?” Phys. Teach. 47, 456–459 (Oct. 2009). See supplementary material at doi.dox.org.xxx for the model template and sample group work questions. Kelly Krieble and Joseph L. Powlette, “A simple apparatus for optical polarization experiments,” Phys. Teach. 41, 537–541 (Dec. 2003). A detailed description of this activity can be found in the Exploritorium Science Snackbook: Cook Up Over 100 Hands-on Exhibits from Everyday Materials (Jossey-Bass, 1009)

Fig. 2. When the two components shift by p/2 rad, the beam is circularly polarized. It is easiest to see the circularity of the polarization if you follow the direction of adjacent peaks from one end of the model to the other.

Supplementary Material “Representing Circular Polarization with a Box of Cereal,” Eugene Torigoe Sample collaborative group work questions: 1) A half-wave plate is just like a quarter-wave plate, but it shifts the components by half a wavelength instead of a quarter wavelength. What would happen to the linearly polarized beam if it travelled through a half wave plate? 2) Elliptically polarized light can be achieved if one component is larger than the other component. Starting with the same beam and a quarter-wave plate, how could you create elliptically polarized beam? 3) Describe the motion of the electric field if the components shifted only 1/8 of a wavelength. 4) What would happen to the intensity of the circularly polarized beam if it were sent though a linear polarizer? How does the final intensity change if the linear polarizer is rotated? 5) There are two types of polarization that describe the orientation of rotation: right circular polarization and left circular polarization. If you point your thumb from your right (left) hand in the direction of propagation, your fingers will curl in the orientation of right (left) circular polarization. If you get right circularly polarized light after sending the beam through a quarter-wave plate, then what would you change to get left circularly polarized light? Model template: Glue or print the template to cardstock paper and cut it out. If you do not have cardstock paper, then cardboard from a box of cereal also works very well. Align the two points marked “a” and slide the two pieces together to create the model (this may take some finesse). Add tape to the diagonal cuts to prevent the pieces from disconnecting. The two pieces should easily slide pass one another.

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THE PHYSICS TEACHER ◆ Vol. 50, March 2012