### Introduction

The Faraday effect or Faraday rotation is a magneto-optical phenomenon — that is, an interaction between light and a magnetic field in a medium. The Faraday effect causes a rotation of the plane of polarization which is linearly proportional to the component of the magnetic field in the direction of propagation.
This effect occurs in most optically transparent dielectric materials (including liquids) under the influence of magnetic fields.
The Faraday effect is caused by left and right circularly polarized waves propagating at slightly different speeds, a property known as circular birefringence. Since a linear polarization can be decomposed into the superposition of two equal-amplitude circularly polarized components of opposite handedness and different phase, the effect of a relative phase shift, induced by the Faraday effect, is to rotate the orientation of a wave’s linear polarization.

The image below shows the faraday effect, B is the magnetic field, while β is the rotation angle of the plane of polarization.

E = ER + EL : the electric field vector of the linear polarized light wave can be thought as the superposition of two circularly polarized vectors, (right) ER and (left) EL .

A magnetic field causes the Zeeman Effect, which consists of a shift of energy levels, which in turn causes the phenomenon of birefringence, that is the speeds of the two circularly polarized waves are different : nR ≠ nL.
You can easily prove that the rotation angle is calculated with the following equation :

β = π * (ν/c) * d * (nR – nL)

Where ν is the frequency, c the speed of light, d the length of the path.
It turns out that the the difference between the refraction indexes of the two waves is proportional to the magnetic field. Then also the rotation angle is proportional to the magnetic field.

β = Cv * d * B

Where the proportional constant Cv is called Verdet constant.

### Physical Interpretation

The linear polarized light that is seen to rotate in the Faraday effect can be seen as consisting of the superposition of a right and a left-circularly polarized beam.

In circularly polarized light the direction of the electric field rotates at the frequency of the light, either clockwise or counter-clockwise. In a material, this electric field causes a force on the charged particles (electrons) comprising the material. The motion thus effected will be circular, and circularly moving charges will create their own magnetic field in addition to the external magnetic field. There will thus be two different cases : the created field will be parallel to the external field for one (circular) polarization, and in the opposing direction for the other polarization direction – thus the net B field is enhanced in one direction and diminished in the opposite direction. This changes the dynamics of the interaction for each beam and one of the beams will be slowed down more than the other, causing a phase difference between the left and right-polarized beam. When the two beams are added after this phase shift, the result is again a linearly polarized beam, but with a rotation in the polarization direction.

The direction and the intensity of polarization rotation depends on the physical properties of the material.

### The Equipment

In order to obtain a measurable Faraday effect it is necessary to have relatively intense axial magnetic fields. One of the simplest ways is to use a solenoid and insert the sample into the solenoid itself. The images below show the solenoid we used and one of the phases of the winding work. The end result is a solenoid of about 1000 turns with a length of 11 cm. The enamelled copper wire has a diameter of 1 mm (18-20 AWG). The “generous” diameter allows us to use relatively high currents (2-4 A). The solenoid thus obtained has resistance R = 3.2 Ω and inductance L = 3.7 mH.

As can be seen in detail above, the solenoid is equipped with a free wheel (flyback) diode, to be connected in parallel to the solenoid, in the case of on / off operation. A resistance (value 1Ω) has also been provided, to be connected in series, in order to measure, through the voltage drop, the current circulating in the solenoid.

Inside the solenoid there is the place to insert the sample subjected to measurement. For this purpose we used an aluminum tube, cut to size, with a length of 13 cm, closed at both ends with plexiglass disks. The plexiglass disks have been fixed with cyanoacrylate glue which also guarantees the seal to the liquids that are introduced (with a syringe) through the side hole. The image below shows the sample holder tube.

The solenoid can be DC driven, in order to evaluate the polarization of the outgoing beam as a function of the intensity of the magnetic field generated by the solenoid ; or it can be AC driven with a sinusoidal signal so as to produce a variable magnetic field and measure at output the variable luminous intensity produced by the magnetic field variations.
For weak signals, such as that induced by the Faraday effect, the measurement of time-varying signals, rather than continuous values, can improve the signal-to-noise ratio. The sinusoidal signal is obtained starting from a function generator connected to a mono audio amplifier, which, in turn, drives the solenoid. The image below shows the amplifier used.

The setup includes the He-Ne laser (Laser He-Ne (Eng)), the polarizing filter, the analyzer filter mounted on a rotating support (Polarization of Light) and the photometer for measuring the intensity of the transmitted light (PSoC based Photometer); all placed on our optical table.
The image below shows the setup with all the components.

### The Experiment

In order to maximize the variation in light intensity read by the photometer, following the application of the magnetic field to the sample under examination, it is convenient to rotate the analyzer polarizer at an angle equal to π/4 (45°).
We know that the intensity transmitted by the analyzer is equal to It = I0 cos2φ.
Differentiating with respect to φ we get :

dI/dφ = 2I0 x cosφ x senφ = I0 x 2 x cosφ x senφ = I0 x sen2φ => Max per φ = π/4

As a substance we used demineralized water that was inserted into the sample holder tube using a syringe. The hole was closed with adhesive tape and the tube was inserted inside the solenoid, as shown in the images.

The first part of the experiment was done by measuring the light intensity of the beam transmitted through the polarizers and the sample under examination, in the absence of current and then with a continuous increasing current in the solenoid. The measurements were made with current equal to : 0, 2, 3, 4 and 4.8 Ampere. For each of these values ​​the variation in light intensity was measured.
The result is shown in the following graph.

For small magnetic field values ​​(such as those generated by our solenoid) the variation of the polarization angle is small therefore we can consider constant the ratio dI/dφ = I0 x sen2φ, in these conditions the variation in transmitted light intensity is proportional to the variation of the polarization angle, which, in turn, is proportional to the magnetic field and therefore to the current in the solenoid. The graph well represents the almost linear link between current and variation in light intensity.

In the second part of the experiment we drove the solenoid with a low frequency sinusoidal signal (for example 100Hz). The signal is obtained from a normal signal generator and then amplified by a normal 100W mono commercial audio amplifier (in practice the solenoid is seen as a speaker). In series with the solenoid there is a resistance of 1Ω at the ends of which the signal that is used as a direct indication of the magnetization current of the solenoid is taken. This reference signal is displayed on the oscilloscope together with the signal produced by the photometer. The image below shows the setup.

As the magnetization current varies, we expect a consequent variation of the signal read by the photometer. Since these are time-varying signals we can use the oscilloscope in AC mode, this allows us to appreciate even small amplitude signals.
The images below show the signals on the oscilloscope: in yellow the trace corresponding to the magnetization current in the solenoid and in blue the trend of the light intensity transmitted by the analyzer. We can see how the trend of the photometer signal is synchronized and proportional to that of the current.

### Conclusions

The experiment carried out allowed us to study the phenomenon of birefringence induced by the magnetic field: Faraday effect. Measurements have been made only in terms of quality. To make more accurate evaluations it is necessary to use a higher quality solenoid in order to know precisely the magnetic field produced. The use of variable signals also allows us to have greater sensitivity and potentially gives us the possibility of adopting sophisticated signal detection techniques such as the Lock-in amplifier.

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