OP16 - The Faraday Effect

Richard Mann, Oriel College

1. Abstract

This is a report of the experiment OP16 of the Oxford Physics Practical Course (The Faraday Effect). The experiment was carried out by Richard Mann and Clive Kirk, and the date recorded in the experimental log is 22/10/02. This report describes the details of the experiment, the results obtained, comparisons with theoretical models and the conclusions drawn from the experiment.

2. Introduction

The Faraday effect is an important tool in modern optical techniques. The phenomenon is a special case result of the Zeeman effect, which alters the properties of dielectric in an applied magnetic field by shifting the energy levels of the constituent atoms. The observational aspect of the Faraday effect can be summarised as the rotation of the plane of polarisation of a beam of plane polarised radiation travelling through the dielectric parallel to the direction of the applied field.

The effect is found to depend upon the wavelength of the incident radiation. The object of this experiment was to determine experimental values of the magnitude of the effect, as parameterised by the Verdet constant (see section 3.5), and to compare them to the theoretical predictions of a simple model based on a harmonic oscillator treatment of the electrons in the dielectric.

3. The Theoretical Model

As stated in the introduction the theoretical model used for comparison with the experimental results is based on a harmonic oscillator approximation for the behaviour of the electrons in the dielectric. Whilst the entirety of that treatment is given in the course manuscript (appendix A, section 2), it is appropriate to overview that treatment here, both to expand on the physical reasoning behind the theory and also to cover areas of the model that were left for the student to determine.

3.1. The Zeeman Effect

A description and derivation of the Zeeman effect can be found in any book on atomic physics, and it is not the place of this report to derive or detail the effect in its entirety. For the purposes of this experiment the important aspect of the effect is that when viewed along the field direction (as by definition is the case in the Faraday effect) a single spectral line of angular frequency ω₀ splits into two circularly polarised components of frequency ω₀ ± eB/2m. This is the case for emission and using the harmonic oscillator assumption we can interpret these as the natural or resonant frequencies of the electrons when circularly polarised light is incident.

3.2. The Equations of Motion for the Harmonic Oscillator

From simple Newtonian Mechanics and the force on an charged particle with negligible magnetic field being F = qE, we obtain the equation of motion of an electron in the dielectric before the magnetic field is applied.

(1)

Where ω₀ is the pre-Zeeman natural frequency of the electron, ω is the frequency of the incident radiation, and E₀ is the amplitude of the radiation. Damping has been neglected as we expect to be using driving frequencies far from the resonant response.

The solution to this equation is simple and given in the manuscript (Appendix A equation (3))

This solution leads to a theoretical value for the refractive index of the material, n, if we assume the permeability to be unity, given by

(2)

With N as the atomic number density.

However, because of the Zeeman shift described in 3.1 the natural frequency of the electrons in no longer ω₀ but ω₀ ± eB/2m depending on the direction of rotation of the incident circularly polarised light. This new resonant frequency replaces ω₀ in equation (2) and thus we can see that n is different for the two alternative circular polarisations. To demonstrate the shift below is a reproduction of figure 1 of the manuscript

Figure 1 - The response of the refractive index to an applied magnetic field

As can be seen from the diagram the function n(ω) shifts in opposite directions for positively rotating and negatively rotating polarisation, thus one has two values of n, n^- and n⁺ for any given value of ω. This leads to the next step in the model.

3.3 Phase Shift

Now that we have determined that the alternative polarisations have different n's we can conclude that they must travel at different velocities v = c/n. This gives rise to a phase shift

(3)

If we take our original plane polarised beam and resolve it into 2 counter rotating circularly polarised components, we can use this phase shift to show that recombining the two components after they have passed through the dielectric the plane of polarisation has rotated by

(4)

3.4. Applying the Model

Before we undertook this experiment the refractive index of the glass being used as the dielectric had already been measured at different frequencies. The data for this is shown below, the first two columns being from the manuscript and the third calculated by us.

Figure 2 - Pre-measured values of n(λ) for the glass dielectric

We used this data to find the value of ω₀ for the glass to use in our model. Using equation (2)

Rearrange to get:

(5)

Which leads to:

(6)

Where n₁, n₂, ω₁ and ω₂ are any two pairs of values from figure 2.

Applying equation (6) to the values in figure 2 we can obtain three values of ω₀

Figure 3 - Theoretical values of ω₀ using pre-measured values of n(λ)

These calculated values give an average result with statistical error:

ω₀ = (1.103 ± 0.019) × 10¹⁶ s^-1

λ₀ = (170.9 ± 2.9) nm

Which puts the resonant response in the UV part of the spectrum. This is far from the frequencies we shall be using, which justifies the no damping approximation in equation (1)

With this calculated value of ω₀ we can also calculate an estimate of N, the atomic number density, using equation (2). Again we can calculate separate values for each pair of n and ω.

Figure 4 - Values of N from calculated value of ω₀ and known values n(λ)

These values give an average value for N with statistical error:

(8.53 ± 0.01) × 10²⁸

3.5. Theoretical Predictions

It is expected that the Faraday effect should be linear with respect to both the applied magnetic field and the length of the dielectric, so we hypothesise that:

θ = VBl

Where θ is the angle of rotation, B is the applied field, l is the length, and V is the Verdet constant defined by this equation, which in general will be a function of λ.

Using the calculated values of N and ω₀ we can use our model to predict values for the Verdet constant V(λ), and to predict the relative magnitudes of the Faraday effect for different frequencies of incident radiation Using the approach suggested in the manuscript we can differentiate equation (5) with respect to ω₀ giving:

which leads to:

and we can associate Δω₀ with the Zeeman shift eB/2m discussed earlier.

Using equation (2) with our calculated values we can predict a value for n(λ) at any λ. We intended to make a comparison between a predicted and an observed ratio of the size of the Faraday effect rotation at λ = 546nm (mercury green) and λ = 589nm (sodium yellow). While we were given the value of n for the latter, the former had to be derived from equation (2), being determined to be n₅₄₆ = 1.937396

From the earlier theory we have:

(7)

Equation (7) gives us the following values:

(n⁺ - n^-)₅₄₆ = 2.6499 × 10^-6

(n⁺ - n^-)₅₈₉ = 2.585 × 10^-6

Using equation (4) it immediately follows that:

θ₅₄₆/θ₅₈₉ = 1.106

We can also estimate the Verdet constant V(λ=546nm), recall that V is defined by:

θ = VBl

V = θ/Bl

From equations (4) and (7)

Which, using the calculated values of ω₀ and n gives V(546) = 152.5 rad m^-1 T^-1

4. The Experiment

4.1. Experimental Technique

The Following diagram is a reproduction of Figure 2 of the manuscript. It shows the main components of the experiment.

Figure 5 - The experimental setup

From the left one can see the lamp (either mercury or sodium), followed by a condensor lens. Then with have polarising prisms to create the plane polarised incident beam. These polarising prisms are a large prism followed by a small Nicol prism covering half the view to create and incident beam that is polarised in 2 orthogonal planes over 2 halves of the view. Powered from the power supply there is a solenoid to generate the applied magnetic field, and inside is the glass sample being used as a dielectric. The outgoing beam then enters a telescope with a rotatable analysing polariser inside.

The reason for using the two prism arrangement is one of accuracy. To find the angle through which the polarisation plane had moved we could find the point of extinction with and without the field, and subtract one from the other. However, it is difficult to judge the point of extinction well as the change in intensity with angle is zero in the first order approximation, i.e dI/dφ = 0. Hence small changes in angle around φ_extinction will not produce a noticable change in intensity, making the point itself hard to judge. Instead we choose to find the point of equal intensities with the 2 polarisation arrangement, where the effect has a first order effect and thus is far more precise. We also choose the point of equal intensities near to extinction rather than close to maximum transmission as this is also easier for the human eye to judge.

4.2. Results and Analysis

First the position of the point of equal intensities was measured with respect to the angular measure on the rotating analyser in the absence of any field. 5 independent measurements were made for φ

φ₁ = 143.5^o, φ₂ = 143.9^o, φ₃ = 143.8^o, φ₄ = 143.5^o, φ₅ = 143.2^o

Giving φ = (143.6 ± 0.3)°

Then the change in φ, Δφ, was measured against the current supplied to the field generating solenoid, using first the mercury green spectral line (546nm), giving the following results

Figure 6 - Rotation of plane of polarisation with supplied current (λ = 546nm)

This was originally plotted on the Practical Course computers (the original plots can be found in the log) to supply a gradient for the expected straight line fit, and the statistical error. A new plot is shown below, but statistical figures remain those from the original plots.

Figure 7 - Plot of rotation against supply current (λ = 546nm)

As can be seen the expectation of a linear fit is well approximated by the data. The statistics from the original plots give:

Δφ = (2.20 ± 0.06)I in degrees (8)

Appendix A1 of the manuscript demonstrates how to calculate the field inside the solenoid in terms of the current given the geometrical shape of the solenoid. Applying this (8) becomes:

Δφ = (1.68 ± 0.05)B in radians (9)

For completeness, with reference to Appendix A1 of the manuscript, the values measured and used were, l = 20cm, secφ₁ = 1.0032, secφ₂ = 1.25.

Recalling the definition of the Verdet constant

V = Δφ/lB, with the measured value of l being 5.08cm

Gives V(546) = 33.1 rad m^-1 T^-1

The entire preceeding experiment was then repeated using the sodium yellow line (589nm). The table of measured Δφ against supplied current is shown below.

Figure 8 - Rotation of plane of polarisation with supplied current (λ = 589nm)

As with the mercury green case, the statistics of these results were obtained from a linear fit plot on the Practical Course computers, which can be found in the log. A new plot is shown below to demonstrate the data only.

Figure 9 - A plot of rotation against supplied current (λ = 589nm)

The original plot gives statistics such that:

Δφ = (2.12± 0.09)I in degrees (10)

which by Appendix A of the manuscript gives

Δφ = (1.62 ± 0.01)B in radians (11)

which by V=Δφ/lB gives V(589) = 31.9rad m^-1 T^-1

The ratio between the effects at 546nm and 589nm as predicted in section 3.5 is:

Δφ₅₄₆/Δφ₅₈₉ = V(546)/V(589) = 33.1/31.9 = 1.04

This compares to the predicted value from section 3.5 of:

θ₅₄₆/θ₅₈₉ = 1.106

4.2.1. Summary of Results

The Verdet constant was found to be:

V(λ = 546nm) = 33.1 rad m^-1 T^-1

V(λ = 589nm) = 31.9 rad m^-1 T^-1

And the ratio of these two, which gives the ratio of the two effects is:

V(546)/V(589) = θ₅₄₆/θ₅₈₉ = 1.04

The experimental value of V(546) compares poorly to the predicted value from section 3.5 of 152.5 rad m^-1 T^-1. However, the experimental value for V(589) is close to V(546) which suggests that the error is systematic rather than statistical. This is most likely to be due to several large assumptions made in the theoretical model. For example, the harmonic oscillator assumption is not very good for glass, as its non-crystalline structure prevents a purely harmonic response. The assumption that the elctrons are independent of each other is also a possible source of significant error.

A better match is found between the predicted value of θ₅₄₆/θ₅₈₉, which is likely to be because the same systematic error is found in each experimental θ, thus cancelling out in the ratio.

The model does however provide a semi-quantitive way of understanding the effect and predicting its magnitude, as the results are less than an order of magnitude apart. This suggests that the basic principles of the physics involved are much as the model suggests.