Measuring the Faraday effect in olive oil using permanent magnets and Malus' law

The Faraday effect (see e.g. [1]) has a wide range of applications including filtering [2, 3] and limiting feedback in optical systems [4]; it also affects light propagating through interstellar media [5]. The effect occurs when a magnetic field parallel to the direction of light propagation induces a circular birefringence in a medium which causes linearly polarized light to rotate as it travels through the medium. The magnitude of this rotation (θ) may be expressed as θ = VBl, where B is the magnetic field strength, l the path length and V is a material-dependent factor known as the Verdet coefficient [1].

Studying the Faraday effect (see e.g. [6, 7]) and other examples of circular birefringence such as optical rotation in sugar solutions [8–10] is a common experiment in undergraduate teaching labs, as it elegantly links the concepts of polarization, dichroism, birefrigence and scattering [1]. However, measuring the Faraday effect can be challenging, especially in cases where the Verdet coefficient is small or the optical path length is short. There are many different methods proposed in the literature, some utilizing DC fields [11–15] and some AC fields [16–22]; by modulating the field and using a lock-in amplifier it is possible to extract rotation angles in the range of microradians. Most experiments employ electromagnets producing fields typically in the range of up to 0.025 T [16, 21] although experiments with larger fields have been performed [12].

In this paper, we report on a simple Faraday measurement apparatus based on small permanent magnets where fields over 0.6 T are easily accessible. The experiment is inexpensive as it does not require the use of an electromagnet or lock-in amplifier. We show that by fitting a Malus' law transmission curve it is possible to measure polarization rotations as small as 50 μrads without the need for field modulation. Consequently, the DC Faraday effect is observable even in weakly magneto-optical materials.

To demonstrate the capabilities of our apparatus, we have measured the Verdet coefficient of olive oil, which was of interest due to a suggestion in the literature of an anomalously high Verdet coefficient at 650 nm [23], contradicting more recent measurements [21]. Our measurements show that the Verdet coefficient of olive oil is similar to other liquids such as water and much smaller (about 2 rad T⁻¹ m⁻¹ at 780 nm) than Faraday active magneto-optical materials such as the crystal TGG (82 rad T⁻¹ m⁻¹) and resonant media such as Rb vapour (1.4 × 10³ rad T⁻¹ m⁻¹) [4].

From Malus' law (see [1]) the optical power (P) of a of linearly polarized light passing through a polarizer is

$$\begin{eqnarray}&&P={P}_{0}{\cos }^{2}\phi ,\end{eqnarray} \tag{ 1 }$$

where P₀ is the incident power and ϕ is the angle of the polarizer relative to the polarization direction of the incident light. Light passing through a circularly birefringent medium has its plane of polarization rotated by an angle θ. When the optically active medium is placed between the polarizers, we may describe the light transmitted through the polarizers using

$$\begin{eqnarray}&&P={P}_{0}{\cos }^{2}(\phi +\theta )+c.\end{eqnarray} \tag{ 2 }$$

Here we have introduced an offset c to account for imperfect extinction of the polarizers and the presence of background light. It is worth noting that as the laser beam has a fixed size, the optical power is linearly proportional to the optical intensity (i.e. the power per unit area) by

$$\begin{eqnarray}&&{I}_{0}=\displaystyle \frac{{P}_{0}}{\pi {w}_{0}^{2}/2},\end{eqnarray} \tag{ 3 }$$

where I₀ is the incident intensity and w₀ is the laser beam radius. Therefore if desired it is possible to frame (1) and (2) in terms of optical intensity rather than optical power. The magnitude of rotation for a given magnetic field strength (B) and path length (dl) is material-dependent and classified by the Verdet coefficient (V) such that

$$\begin{eqnarray}&&\theta =V{\int }_{0}^{l}B{\rm{d}}l={{VB}}_{\mathrm{av}}l,\end{eqnarray} \tag{ 4 }$$

where B_av represents the average magnetic field strength across the sample and l is the sample's total length. Due to its relation to the material's refractive index [15], the Verdet coefficient is found to depend on the temperature and wavelength so may be modelled using a dispersion law. We tested this by determining the Verdet coefficient at multiple wavelengths and attempting to fit a dispersion curve. We compared the data to two separate models, the first being a Cauchy-type [24, 25] dispersion of the form

$$\begin{eqnarray}&&V(\lambda )=A+\displaystyle \frac{B}{{\lambda }^{2}},\end{eqnarray} \tag{ 5 }$$

where A and B are both fitting parameters and λ is the wavelength of light. Secondly we attempted to fit a Drude-type [26] dispersion of the form

$$\begin{eqnarray}&&V(\lambda )=\displaystyle \frac{A}{{\lambda }^{2}-{\lambda }_{0}^{2}},\end{eqnarray} \tag{ 6 }$$

where now A and λ₀ are the two fitting parameters. We believed using the Drude model would provide a more profound insight into the underlying physics as the value of λ₀ should correspond to an absorbance peak.

The experimental apparatus consisted of simple optical equipment, commonly available in an undergraduate laboratory. The arrangement of the equipment is shown in figure 1. Measurements of optical rotation as a function of the magnetic field strength were obtained at seven different wavelengths (405.4 nm, 446.6 nm, 518.8 nm, 637.8 nm, 659.2 nm, 681.8 nm and 796.2 nm) using HEXA-BEAM lasers (Photonics Technologies). Five of the wavelengths were studied using one HEXA-BEAM laser. This device was then replaced by another HEXA-BEAM laser and the experiment was realigned to study the final two wavelengths. The advantage of using the HEXA-BEAM lasers is—depending on the modules installed—up to six wavelengths may be studied using a single device, without the need for realignment.

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Light from the laser passes through a linear polarizer to ensure it is fully polarized before entering the sample. Polarized light then passes through a glass cuvette (Thorlabs CV10 Q3500F) containing an olive oil sample (Tesco Olive Oil) inside a magnetic field, and then through to an analyser which consists of a second polarizer mounted in a rotation stage (Thorlabs PRM1/MZ8). Finally, the light is incident on a photodiode (Thorlabs PDA100A-EC) which measures the transmitted optical power (P in (2)). The photodiode is connected to a data acquisition device (National Instruments USB-6008) so that the optical power may be measured automatically on a PC.

The magnetic field was provided by a pair of neodymium (NdFeB) magnets in an aluminium mount; a schematic view of the magnets may be seen in figure 1 (b). The magnetic field strength could be varied by changing the distance between the magnets. The threaded design of the magnet holders and mount allowed the separation to be adjusted by rotating the magnet holders. Using a Hall probe, the magnetic field was measured at various magnet separation distances to determine its spatial homogeneity. Further details of the construction of the mount for the magnets and the full magnetic field strength calibration may be found in appendix A. The same magnets may be used in a wide range of different applications (see for example [27]).

The rotation stage and photodiode were connected to a PC so that the angle of the analyser could be controlled and the optical power at the photodiode (P) could be measured automatically. A computer algorithm was programmed which rotated the analyser in increments of one degree and then measured the optical power of the light using the photodiode. In this way, a mean power with a standard error was measured at each angle over a full 360° analyser rotation. The data were then fitted using Malus' law, as described in (2). The major advantage of using computer-controlled apparatus was the rapid collection of data—a full 360° rotation cycle was collected in approximately 10 min.

Applying the Malus' law fit to the data, as demonstrated in figure 2, yielded a value of the rotation (θ). Many experiments in the literature will measure rotations by observing the power change at a fixed analyser angle—often 45°—where the power change will be maximized. However, the residuals of figure 2 indicate that measurements performed at a single angle are sensitive to short-term changes in the incident power and the offset, caused by power fluctuations of the laser and changes in background levels of light. Our method of fitting the model to the full range of the data reduces this issue by fully parameterizing the amplitude and offset. Hence, we may extract rotations precisely, insensitive to random short-term power fluctuations which occur within a single measurement. However, a limitation of our method is that we cannot account for fluctuations over a larger timescale such as a long-term drift in the laser power or background light level. As such, care must still be taken to control these parameters whilst acquiring data.

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For each sample being measured, three sets of data were taken at each magnetic field strength to obtain a mean rotation with a standard error. The field strength was then altered and the new value of θ was calculated using the same procedure. At each magnetic field strength, we compared the measured rotation to the rotation obtained at the minimum field strength (215.4 ± 1.4 mT). Here we introduce some new notation: Δθ = θ − θ₀, where θ₀ is the rotation at the lowest field value and Δθ is the change in rotation. Ideally, the rotations would be compared to the sample in the absence of any field but this was not possible as the cuvette had to remain fixed in place during measurements due to the natural birefringence of the glass (this was achieved using a custom 3D-printed holder which slotted the cuvette in place inside the magnetic field). However, measuring from zero field would only introduce an offset to the measurements and not affect the gradient from which the Verdet coefficient is determined.

Figure 2 shows example data that have been fit with a weighted least squares algorithm using the Malus' law method according to (2). For the data shown, the rotation change Δθ = 2.19° ± 0.05°. This is one of the highest values that was observed and emphasizes the need for precise measurements.

When measuring the Verdet coefficient of olive oil, the glass cuvette in which the olive oil is contained also experiences the Faraday effect and so induces extra optical rotation of the transmitted light. This may be seen in figure 3 where it is clear that the glass makes a significant contribution to the rotation. To account for this, measurements were made of the optical rotation due to the empty glass cuvette. The optical rotation due to the olive oil can then be calculated from a 'combined' measurement of the optical rotation of the cuvette full of oil using

$$\begin{eqnarray}&&{\rm{\Delta }}{\theta }_{\mathrm{oil}}={\rm{\Delta }}{\theta }_{\mathrm{combined}}-{\rm{\Delta }}{\theta }_{\mathrm{cuvette}},\end{eqnarray} \tag{ 7 }$$

and so the Verdet coefficient may be determined using

$$\begin{eqnarray}&&V=\displaystyle \frac{{\rm{\Delta }}{\theta }_{\mathrm{oil}}}{{B}_{\mathrm{oil}}{l}_{\mathrm{oil}}},\end{eqnarray} \tag{ 8 }$$

where B_oil is the average field strength experienced by the oil and l_oil is the path length of the oil. As this equation implies, a plot of Δθ_oil versus B_oil allows the Verdet coefficient to be calculated by dividing the gradient by the path length.

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As described in section 3, the Malus' law fitting procedure was repeated to obtain Δθ_oil values at each field strength such that a linear fit could be performed. The linear fits for 4 of the wavelengths are displayed in figure 4. Because there were errors in B_oil values and in Δθ_oil, the data were fit using an orthogonal distance regression algorithm. The errors on the fitting parameters were obtained from the square roots of the relevant entries in the covariance matrix (procedure as outlined in [28]). The uncertainty in the Verdet coefficient was then propagated from the error in the gradient and the error in the path length. Table 1 gives the measured Verdet coefficients and uncertainties at each wavelength.

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As shown in figure 5, the measured Verdet coefficients were fit with Cauchy-type and Drude-type dispersion curves using (5) and (6) respectively. For the Cauchy-type fit, the parameters were found to be A = −26 ± 4 deg T⁻¹ m⁻¹ and B = 9.42 ± 0.12 × 10⁷ deg T⁻¹ m⁻¹ nm² while for the Drude-type fit they were A = 7.9 ± 0.2 × 10⁷ deg T⁻¹ m⁻¹ nm² and λ₀ = 142 ± 13 nm. As before, the errors on these fitting parameters were obtained from the square root of the relevant entries in the covariance matrix.

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The dispersion relationship we determined is similar to that of other liquids such as water [16]. Our determined Verdet coefficients are also in good agreement with recent values present in the literature at similar wavelengths [21]. Similar to these recent measurements, we did not observe an anomalously high Verdet coefficient at 650 nm as was previously reported [23].

A restriction of using our permanent magnet set-up is that much smaller path lengths must be used than, for example, in a solenoid set-up. However, the high field strengths we were able to achieve help compensate for this and so measurable rotations are still easily observed. The path length of the oil was measured precisely using digital calipers so despite being small it did not introduce major uncertainty into the Verdet coefficient values.

We believe the major strength of our method is the rigorous Malus' law fitting which allows us to average over short-term random fluctuations in the laser power and detect rotations with errors on the order of tens of μrads. Performing the fit and allowing for an offset to be paramaterized makes the method insensitive to short-term fluctuations in the background level of light which are often difficult to control, particularly in a teaching laboratory. However, as discussed previously, our method is insufficient if there are significant long-term drifts in laser power or background light levels. If this were found to be a major issue, one possible way to eliminate it could be to introduce a second photodiode to measure a blank background signal which may then be subtracted from the measured signal.

Traditionally, measuring the transmitted optical power over a full range of angles would be a tedious process, requiring manual rotation of the analyser. Using the computer-controlled photodiode and analyser makes the data collection much easier and faster. This means a Verdet coefficient may be determined even in a short laboratory session. The coding of the algorithm is also a vital skill as in modern physics laboratories data collection is becoming increasingly automated. The coding can either be partially or fully completed by students to give them an understanding of how the computer is interfacing with the equipment.

As shown in figure 2, the raw data give an excellent visual demonstration of Malus' law which students should be familiar with from earlier levels of teaching. Performing the fit provides students with a good opportunity to develop their model fitting and error analysis skills.

Experimental set-ups to study the Faraday effect are available commercially (see for example [29]). In comparison, our set-up based on permanent magnets, allows open access to the higher fields enabling the investigation of a wide range of weakly magneto-optical materials, including liquids. In addition, using the HEXA-BEAM laser, up to six wavelengths may be studied quickly rather than just one, making it easier to observe dispersion effects. Perhaps the greatest advantage of our set-up over commercial system is the automation using the rotating polarizer and data acquisition device which, as stated, makes measurements easier, faster and more precise than manual operation.

Depending on the level of students, many aspects of the experiment may be partially or fully completed for them, to tailor the learning to their skill set. For instance, if they are not expected to be able to perform non-linear fits, they could be provided with the Malus' law fitting algorithm. In particular, the full process of calibrating the magnetic field, as set out in appendix A, was fairly advanced and for students it should be sufficient to determine B_oil at each magnet separation simply by taking an average of several measurements. The full calibration then presents an interesting area in which students can extend the experiment. Other avenues for extension include attempting to use the Verdet coefficient to detect adulteration of olive oil (similar to another experiment in the literature [30]) or measuring the Verdet coefficient of other fluids.

We have devised a simple experimental procedure based on Malus' law which measures polarization rotations with a precision of up to 50 μrads. The versatility of the Malus' law fitting method allows it to be used in a variety of different polarimetry experiments including the study of chirality in sugars, stress-induced birefringence in plastics and magneto-induced birefringence in liquids and gases. To prove the efficacy of our method, we studied the Faraday effect in olive oil and were able to measure Verdet coefficients with errors as small as 0.4%. Our results agree well with values in the literature and we present excellent fits to dispersive models. The experiment is simple in its execution and equipment, yet can provide profound insight into fundamental concepts in optics and electromagnetism and as such is well-suited to being used as an undergraduate teaching experiment. The experiment provides training in a wide range of experimental skills, including data analysis and error analysis. In addition, the versatility of the apparatus provides ample opportunity for extension.

We would like to thank Dr Aidan Hindmarch and Dr Jason Anderson for providing data aquisition software, Chloe So for help and advice, Durham University for providing the equipment and funding required to complete this investigation and the reviewers for their time and helpful input.

NLRS and CSA acknowledge financial support from EPSRC Grant No. EP/M014398/1. IGH and CSA acknowledge financial support from EPSRC Grant No. EP/R002061/1. The data in this paper are available from [31].

As described previously, the magnetic field was provided by a pair of annular neodymium permanent magnets. The magnets were N52 grade, with a NiCuNi coating and were produced bespoke according to our mechanical drawings (see supplementary material is available at stacks.iop.org/EJP/41/025301/mmedia) by Jinmagnets Industrial Co. (http://jinmagnets.com). Whilst we chose to use custom-produced magnets, any strong annular or ring-like magnet (which are common and cheap) would be suitable.

To hold the magnets in place we machined custom magnet fittings and custom holders. The cylindrical magnet fittings were machined from brass and threaded. The holders were machined from aluminium and similarly threaded allowing the fittings to screw in and making it simple to alter the magnet separation; we machined brass handles for the fittings to make the screwing easier. The magnet holders were kept apart by aluminium separators at the top and bottom. For full technical drawings of the magnets and our custom mount, please see the supplementary material.

The magnetic field was calibrated to determine the spatial homogeneity and average field strength at each magnet separation distance. Each full rotation of the magnets in their mount increased their separation by approximately 2 mm and therefore reduced the magnetic field.

The spatial homogeneity was measured using a transverse Hall probe which was moved along the direction of laser propagation through the magnetic field (z axis). These measurements are summarized in figures A1(a) and (b). Theory curves were fit to the data by considering the axial field strength B_z at a distance z from a cylindrical magnet with uniform magnetization B₀ and radius R:

$$\begin{eqnarray}&&{B}_{z}=\displaystyle \frac{{B}_{0}}{2}\left[\displaystyle \frac{z+{z}_{0}+t}{\sqrt{{\left(z+{z}_{0}+t\right)}^{2}+{R}^{2}}}-\displaystyle \frac{z+{z}_{0}-t}{\sqrt{{\left(z+{z}_{0}-t\right)}^{2}+{R}^{2}}}\right],\end{eqnarray} \tag{ A.1 }$$

where z₀ is an offset to account for the fact that the magnet is not at the origin and t is the thickness of the magnet (for more detail see [32]). As seen in figure 1(b), the magnets were not perfect cylinders and so the field produced by the magnet was determined by piecewise addition of the field due to each part of the magnet and a subtraction due to the hole bored through the magnet's centre. It was then possible to sum the contributions from each magnet to work out the overall field strength at each z position.

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The theory curves were used to calculate the average field strength experienced by the oil (B_oil) for each separation distance; these results are summarized in figure A1(c). It should be noted that the average field strength experienced by the combined sample (B_combined) and cuvette (B_cuvette) are not the same as for the oil alone. However, as shown by (8), B_oil is the only field strength required to calculate the Verdet coefficient of the oil.

As discussed in the results section, to determine the Verdet coefficient for olive oil it was necessary to perform a separate set of measurements with an empty cuvette to subtract the rotation induced by the glass. A side-effect of performing these measurements is that we were able to determine Verdet coefficients for the UV fused quartz glass at each of the wavelengths we tested; these are summarized in table B1. In general, the glass Verdet coefficients have a larger uncertainty than those for the oil, this could be due to the shorter path length of the glass.

Figure B1 shows the dispersion curves applied to our data for the glass. As with the olive oil data, the dispersive models fit well and our data is in good agreement with the literature [33]. The exception is the data point corresponding to 796.2 nm which is far from the models and has a large error. We believe this may be due to the polarizers being unsuited to the near-IR region. Alternatively, the quartz glass may possess an absorption peak in the IR region, such that this wavelength is actually part of a separate dispersion curve—though this would require further study to confirm. Whilst we focused on the Faraday effect in olive oil, we show here that our method may be used more generally and also works well for the Faraday effect in solids.

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