Transcript Slide 1

FARADAY ROTATION
Gennady Voronov
In this experiments we experimentally
determine the Verdet constant of a glass rod with
specification SF-59 to be 0.2375rad/mT*cm. We
observed a rotation in the plane of polarization of a
linearly polarized monochromatic beam of light,
which we obtained by shining a red laser through an
analyzer polaroid. The rotation occurs as the beam
propagates through a dielectric in the presence of a
magnetic field and the magnitude of the rotation is
related to both the Verdet constant and projection of
the magnetic field in the direction propagation of the
laser beam. In this manner we determine the Verdet
constant of SF-59. Finally to achieve a greater
accuracy we model the magnetic field to take into
account that the field is non constant inside the
solenoid. Finally we conduct the experiment once
without a hood and once with a hood to keep out
stray light to check if it makes a difference. We
discovered that the hood did indeed make a
difference.
APPARATUS
The goal of the experiment is to determine the Verdet constant of
transparent dielectric. The experimental apparatus that allows us to determine
the above value consists of 4 basic components: a linearly polarized
monochromatic light source, an analyzer polaroid, a solenoid, and an optical
detector. A linearly polarized monochromatic light source is obtained by shining
a red laser through an analyzer polarizer. The solenoid is 150mm long, and 10
layers. The solenoid is constructed with #18 double insulated copper wire. Finally
the detector is simply a photodiode. The apparatus setup is shown in Fig. 2.
Fig 4. Verdet constant as a function of Avg. B field without hood.
Fig 2. Note the apparatus use specifically in our experiment uses a red laser as a light source and a glass rod for
the dielectric.
PROCEDURE
In order to determine the Verdet constant of material SF-59, we propagate
a laser beam through this material in the presence of a known magnetic field. We
then measure the rotation of the plane of polarization using an analyzer polaroid.
To determine the Verdet constant accurately we do not treat the solenoid as ideal.
By doing so the magnetic field that induces the rotation is non constant.
Equation 1 generalizes as such
The Faraday effect, first observed by Michael
Faraday in 1845, provided the first evidence of a
connection between light and magnetism. Faraday
found that linearly polarized light propagating
through a dielectric parallel to a static magnetic field
had an observable rotation in its plane of
polarization. This is represented pictorially below in
Fig. 1.
  
 B(l )  dl .
glass rod
To determine the Verdet constant, then we must model the magnetic field inside
the solenoid. We do this by treating the solenoid as a series of current loops with
10 radial layers. Then we calculate the field at a point inside the solenoid by
summing over every loop, the contribution to the field from each particular loop.
The field as well as the integral in equation 2 are evaluated numerically. One
additional piece is necessary to compute the magnetic field, the current running
through the solenoid. We can continuously measure the voltage across the
solenoid and via,
The Faraday effect is used in the fabrication of
optical isolators to prevent unwanted backreflections.
The susceptibility of materials and carrier densities
in semiconductors may also be inferred by
measuring the magnitude of the Faraday rotation.
(3)
where V is the voltage, R is the resistance, and I is the current, we may determine
the current if we know R. We measured the voltage and current of across the
solenoid for a few points and then plot voltage as a function of current.
Fig 5. Verdet constant as a function of Avg. B field with hood.
We repeated the measurement with a hood to reduce
stray light entering the photodetector and as we did
for the measurement with a hood we determine a
Verdet constant of 0.2375rad/mT*cm with a variance
of 0.0981rad/mT*cm. The variance with a hood is
greater however that is more likely due to the fact that
the hood measurements were taken after the no hood
measurements, where the measurements were taken
will little time in between and the solenoid over a few
runs did heat up enough for a non-negligible change
in the resistance.
Voltage(Volts) as Function of Current(Amps)
10
Voltage(Volts)
This effect is a result of a magnetic field induced
circular birefringence in the linearly polarized laser
beam in a dielectric material. In a constant magnetic
field the rotation may be quantitatively described by
  Bl
(1)
where Δθ is the rotation angle of the plane of
polarization, ν is the Verdet constant, B is the
magnitude of the magnetic field, and l is the length
of the glass rod. The Verdet constant is a material
property of dielectrics and it is the constant of
proportionality between the angle of rotation and the
magnitude of the magnetic field.
First thing we notice is that the Verdet constant varies
strongly with a weak magnetic field. We think this is a
result of electronic noise in the photodetector which
becomes insignificant as we raise the B field. The
Verdet constant settles to a constant for values of avg.
B field greater than 0.035. We determine the Verdet
constant to be 0.4827rad/mT*cm with a variance of
0.0836rad/mT*cm.
(2)
V  RI
Fig .1 Plane of polarization of light rotated as it propagates through
dielectric in presence of parallel magnetic field.
We plot our results below. First we show the
determined Verdet constant as a function of average B
field over the glass dielectric. Our first set of data was
taken without a hood.
8
6
4
y = 2.5464x + 0.0274
R2 = 0.9994
2
0
0
0.5
1
1.5
2
2.5
3
3.5
Current(Amps)
Fig 3. Measured voltage as a function of current used in determining the resistance of solenoid.
From this we can conclude that the resistance is 2.5464Ω. The solenoid heats up
when current is running through it, increasing the resistance. However we turn
on the current for short enough time intervals so that the increase in resistance is
negligible. Now all we must do is measure voltage across the solenoid and
rotation of the plane of polarization and we may obtain the Verdet constant.
REFRENCES:
Frank J. Loeffler, “A Faraday rotation experiment for undergraduate physics laboratory.” Am. J. Phys. 51 (7) July
1983
Aloke Jain et. al., “A simple experiment for determining Verdet constant using alternating current magnetic
fields.” Am. J. Phys. 67(8) August 1999.
Meyrath, Todd, “Electromagnet Design Basics for Cold Atom Experiments.” Phys. Rev. A 35 November 2004.
We measured a Verdet constant of 0.2375rad/mT*cm
and 0.4827rad/mT*cm with and without a hood
respectively. Clearly using the hood impacts the
determination of the constant. In the future this
experiment could be improved by letting the solenoid
cool in between data runs to insure that the resistance
remains the same throughout the experiment.