2426-12 The Faraday Effect
The University of Hong Kong
Department of Physics
PHYS2426Intermediate Experimental Physics Laboratory
Experiment No. 2426-12: The Faraday Effect
K.F. Ho/Dr. S.W. Fan/Prof. P.K. MacKeown/August 2000
Aim:
The objective of this experiment is to use an optical setup to observe the Faraday Rotation Effect described by equation (1) in flint glass. The wavelength dependence of the Verdet constant in equation (2) is observed as a result of wavelength dispersion in a material medium.
Background:
In 1845 Michael Faraday discovered that the manner in which light propagates through a material medium could be influenced by an externally applied magnetic field. He found that when linearly polarized light passed through a piece of glass, the plane of polarization was rotated in the presence of a strong magnetic field applied in the direction of propagation of the light. The Faraday Effect was one of the earliest indications of the interrelationship between electromagnetism and light Ref. [1].
The effect is used today in applications, and in fundamental physics research. In one application, the Faraday Effect has been used to make optical modulators, Ref. [1]. Such devices can be employed as light switches.
A more current (1999) application of the Faraday Effect is to monitor the extremely fast electron spin precession in semiconductor and semiconductor quantum well materials, Ref [2,3].
Classical Theory of the Faraday Effect
Verdet Constant and Angle of Rotation
The angle , through which the plane-of-vibration of the electric field rotates, is given by the empirically determined expression
(1)
where B is the applied magnetic field (in Tesla), d is the length of the medium traversed (in cm), and V=is a proportionality factor known as the Verdet constant which depends on the wavelength of the light. The unit for V is usually expressed as radian per Tesla meter (1Tesla=10000Gauss). A representative value for V in the case of flint glass is 31700 min of arc/T m (0.0317 min of arc/Gauss cm) from Ref. [1].
This experiment has three basic parts (completion of each depends upon the scheduled time):
(1)Calibration of the magnetic field using a Gauss
meter.
(2) Investigation of the rotation of the plane of polarization, at various settings of the magnetic field, and for a fixed wavelength of light.
(3) Investigation of the rotation of the plane of polarization, at a fixed magnetic field and for different wavelength of light.
Relation to Circular Birefringence
Polarized light passing through the electronic structure of some material media may undergo optical rotation - the phenomenon of circular birefringence. A beam of linearly polarized light can be thought of as a superposition of left and right circularly polarized components with equal amplitudes. For a material that is birefringent, the indices of refraction are different for the left and right circularly polarized components of light passing through the material. Each polarization component traverses the sample with a different refractive index and thus with a different speed. On leaving the sample, left and right circular components are out of phase. The superposition of the two circular polarizations again results in linearly polarized light but with its plane of polarization rotated from its original direction before entering the sample.
Passage of Light Through a Dispersive Medium in a Magnetic Field
In a non-birefringent material, how can an applied magnetic field cause the left and right circularly polarized light to have different indices of refraction? It arises from the interaction of the electrons in the medium with the radiation. However, these electrons are not free, they are bound in the atomic orbits, and when there is a magnetic field applied these orbits precess about the field direction - Larmor precession. As a result from the point of view of the electrons, the two circularly rotating components of the radiation appear to have different frequencies (wavelengths), and if the refractive index depends on the wavelength, , the propagation velocity of the two components will differ. As shown in the appendix this gives rise to a net rotation of the electric vector given by (1), where the constant V is given by
(2)
where is the wavelength of the linearly polarized. This is the expression for the Verdet constant, in units of min arc/(Tesla cm), and it has the value of .
Experiment:
Apparatus
- Transformer Power Supply for lamp (Leybold model 52125)
- 5 Amps Current Supply for Magnets
- Digital multimeter for measurement of magnet current
- A Gauss meter with Hall probe
- Two coils 250 turns for magnets
- Pasco OS 8020 fiber optic light guide and meter display
- Halogen Lamp Housing w/100W lamp (Leybold model 45064)
- Picture Slider and holder for color filters
- Linear polarizer with coarse rotation
- Linear polarizer with fine rotation (1deg resolution)
- Focusing lens with 50mm focal length
- Two linear rails
- Five Adjustable optical mounts
- Rectangular flint glass sample
- Four Color filters
Setup
(see Fig.3)
Procedure:
Halogen Lamp and Current Supply
Before starting the experiment, check that the necessary items are situated with the experimental setup.
(1)Check to see if the halogen lamp is working properly. With the Leybold (model 52125) lamp current supply TURNED OFF, connect two wires from the Leybold supply to the rear connectors of the black halogen lamp unit, which should have already been placed on the optical rail.
(2)On the model 52125 use the left and right most output connectors, since this would give the required output current of 10Amps to the lamp unit. After the electrical connection is made, turn the lamp supply on. If the lamp and/or current supply are working, the lamp should turn on immediately and be very bright. The lamp may need to be stabilized initially for 10 minutes so leave it on for the duration of the experiment.
(Caution: Try not to touch the lamp housing, it can get very HOT very soon. Also never disconnect the current supply wires while the lamp is on. This may cause electrical sparks at the connectors.)
Magnetic Field Calibration
In this part of the experiment, the magnetic field between the two shaped magnet pole pieces is calibrated as a function of the magnet supply current.
(1)First, carefully remove the piece of flint glass placed between the magnet pole faces. Place the piece of flint glass gently into the (black) 35mm film container.
(2)Identify the gauss meter and its accompanied Hall probe. The meter may be a Lakeshore or a UniLab model 612003 gauss meter with a LCD display. Use a clamp stand and clamp to fasten the Hall probe. On the UniLab meter, use the 2000mT range (10Gauss/1mTesla). If the 200mT range is used the displayed result should be divided by 10. On either unit, with the probe fastened, move the clamp stand so that the tip of the Hall probe is situated in the space between the coned pole pieces.
(3)Connect two wires to the two 250 turns coils to a magnet current supply. Make sure that the current supply is turned off before the connections are made. The two coils are situated immediately below the coned magnet pole pieces. The connections between these two coils should be in series. Place a multimeter in series with the coils to monitor the current that is supplied by the magnet current supply.
(4)Turn on the gauss meter and set the meter reading to zero. Then turn on the magnet current supply and slowly increase the current to the magnets. Increment the current in steps of approximately 0.5Amps up to a maximum of 8Amps. For each value of the current, record the reading from the gauss meter.
(5)Make a plot of the measured magnetic field(in Gauss unit) versus the current. This should be a straight line with nearly zero intercept. After the last current value has been measured, slowly, decrease the supply current to the magnets so that the current reduces back to zero.
Focusing Light onto the Fiber Optic Cable
The Pasco OS-8020 Photometer
A Pasco model OS-8020 photometer is used to detect the polarized light. his photometer can measure light level down to a sensitivity of 0.1lux (the level of illuminance on a night with a full moon). The light intensity level, from the polarized beam in a darkened room, is in the range of 0.1-3 lux, and depends on the wavelength of the light.
Alignment of Polarizers
(1)Turn on the halogen lamp by switching on the lamp power supply (Model 52125).
(2)A picture slider is placed after the condenser lens on the lamp housing (see figure 2). The picture slider has two positions and can accommodate up to two color filters. Place just one color filter into placed in the holder. Move the picture slider into a position so that only white light comes out from the lamp.
(3)The beam of white light should pass through the holes of the coned magnet pole pieces. Two linear polarizers should be in place on the optical rail and in the path of the light beam. One is situated between the lamp and the magnet (refer to figure 3 for details). This polarizer is used to form linearly polarized light from the lamp. The second polarizer, called an analyzer, is placed between the magnet and a focusing lens (focal length of 50mm).
(4)Rotate the analyzer polarizer so that maximum brightness can be observed when a piece of paper (or a white screen) is placed just after the analyzer. Remove the piece of paper. At this point, the transmission axes of the two polarizers should be nearly parallel.
Focus of Light Beam
(1)An optical fiber should be mounted, on a stand, as the rightmost component on the optical rail, and the opening in the fiber is placed directly along the beam path. Next, slowly move the f=50mm, focusing lens, along the optical rail so that the halogen lamplight is focused directly onto the entrance of the optical fiber. This may cause the focused light beam to exceed the entrance diameter of the fiber cable. This is acceptable since we need to get as much light into the fiber as possible. Similarly, the brightness of the focused spot can be simultaneously monitored on the Pasco meter display.
(2)To obtain a focused spot, no further adjustment of the lamp would be needed. However, due to repeated usage of the experimental setup, the position of the lamp, along the rail, may change from time to time. If a sharp image of the lamp cannot be made, by simply moving the f=50mm lens along the optical rail, then a focused image may also be obtained by slowly moving the adjustment rod connected to the lamp (see figure 1). To do this, first loosen the arresting screw that holds the adjustment rod. Make sure to lock the arresting screw when a sharp image is formed at the entrance position of optical fiber.
Color Filter and Zeroing of the Light Meter
(1)Place a color filter into one of the empty slots of the slide holder. Place the piece of flint glass back into position between the coned magnet pole pieces. The white light from the lamp should still be able to reach the entrance of the fiber optics cable.
(2)Move the picture slider so that the color filter blocks the white light and only light from the color filter is observed. In a darkened room, one can observe the dim colored light at the entrance of the fiber optics cable.
(Important: The analog meter connected to the optics cable needs to be properly adjusted in order to obtain reliable results.)
(3)Place a piece of paper in front of the optics cable, to block the colored light beam. Now, adjust the zero offset, so that the meter reading is nearly zero. Next, adjust the sensitivity on the meter so that the needle deflects to nearly full scale. For this, the gain on the meter may also need to be adjusted. Remove the piece of paper from the beam path. The meter reading should now be roughly stable and fluctuations of one or two small divisions may be observable.
Figure 1. Pasco OS-8020 photometer. Fiber optics cable not shown.
Figure 2. Leybold model 450 64 Halogen Lamp Housing w/picture slider
Figure 3. The optical fiber from the Pasco OS-8020 photometer (not shown) is placed to the left of the 50mm focal length lens. The optical fiber is mounted so that the focused light beam covers the entrance to the fiber.
(Keep in mind that scattered light from flash lights or the swing arm desk lamp used in nearby experiments may affect the above zero offset adjustment.)
Measurement of the Angle of Rotation
The rotation angles are to be measured for two settings: one is without magnetic field and the other with the magnetic field applied to the flint glass sample. The simplest way to do this would be, with no applied field, to set the polarizer and analyzer axes parallel, i.e. to give maximum intensity, and when the field is applied to find the rotation of the analyzer necessary to restore this maximum.
However, because of the Malus' law, , where is the angle between the plane of polarization of the incident light and the axis of the analyzer, the intensity is least sensitive to the angle at the maximum (, also true at the minimum). The rotation necessary to restore the signal when is a more sensitive measure of the angle of rotation, and it is this method which is described below.
Rotation angle as a function of the applied magnetic field.
With no applied magnetic field.
(1)Slowly rotate the analyzer so as to get a maximum reading, (largest meter deflection), on the light meter. You may need to try locating the maximum by rotating the axis of the polarizer several times. Record the angle value, where this maximum occurs.
(2)Rotate the analyzer to angles on either side of the maximum , such that the measured intensities are 50% of . Record the angle values and designate these as for the angles on the left and right sides of respectively.
(3)Repeat steps (1) and (2) five times and find the average values together with their errors. The error in the average is obtained from the standard deviation of the five measurements.
Use different kinds of color filter, and start magnet supply current for 5 Amps.
The color filters are labeled with serial numbers on them, the wavelength for maximum transmission of the filters are listed below.
(Important: When changing the color filters, handle the filters by the square edges, and try avoid touching the glass portion of the filters.)
Color FiltersWavelength (nm)Serial no.
Violet44046813
Blue w/violet45046811
Blue-green52046809
Yellow57046805
(1)Repeat steps (1) to (3) as in section With no applied magnetic field above, to find the average values .
(2)During the measurement, record any changes in the supply current and include these changes as part of the error for the magnetic field measurement.
(3)The Faraday rotation angle, , is obtained from the difference in the average angle values, , for the two cases with and without applied magnetic field.
(4)Plot the average rotation angle for versus magnetic field.
(5)Repeat the above procedures for magnetic current 6,7 and 8 Amps.
Investigation of the wavelength dispersion of flint glass
(1)Using a fixed value of the magnetic field set the magnet supply current so that the current meter reads 5amps.
(2)Repeat steps (1) to (3) as in section Use different kinds of color filter above, to obtain values of and for the other color filters.
(3)Plot a graph of versus the color filter wavelength.
References:
(1)E. Hecht, Optics 3rd. Edition (Addison Wesley, 1998), p.362.
(2)Lu J. Sham, "Closer to Coherence Control", Science vol. 277, 1258 (1997).
(3)J.M. Kikkawa et al, "Room-Temperature Spin Memory in Two-Dimensional Electron Gases", Science vol. 277, 1284 (1997).
(4)F. L. Pedrotti and P. Bandettini, "Faraday rotation in the undergraduate advanced laboratory", Am. J. Phys. 58 (6) 542 (1990).
(5)F.J. Loeffler, "A Faraday rotation experiment for the undergraduate physics laboratory", Am. J. Phys. 51 (7) 661 (1983).
(6)D.A. Van Baak, "Resonant Faraday rotation as a probe of atomic dispersion", Am. J. Phys. 64 (6) 724 (1996).
(7)RCA Photomultiplier Handbook.
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