# Calculating Magnetic Field Strength from FFT Amplitude

Briefly on the exposition; I'm an undergraduate assistant to a professor at UMass Amherst. We contribute to the Muon g-2 experiment in Fermilab, designing and optimizing the magnetic-measurement equipment. As you might imagine I utilize the Fourier Transform often to analyze data. The data I'm analyzing is photo-current which comes from a laser of known power and is modulated by a magnetic field. The modulation by the magnetic field perturbs the plane of polarization in a well-defined manner (Faraday Rotation), resulting in a modulation of the intensity of photo-current and thus the voltage measured in our DAQ Assist box.

I use LabView to analyze the data and thus use a proprietary "virtual instrument" in the program to do the FFT for me (obviously as the sheer amount of data would be overwhelming!). I understand the concept of an FFT, it being a mathematical tool to take a signal and decompose it into it's frequency domain and measure the strength of each frequency directly by their amplitude, as they contribute to the resultant "summed" signal. However what I need to understand is how to translate the amplitude of one of the frequencies (say 60 hz) into the amplitude for the magnetic field. In other words my professor wants me to be able to tell the magnetic field strength from the amplitudes present on our power spectrum. Now I have a few questions...

i) Can this be done by simply analyzing one frequency and it's amplitude. My intuition says no as I would imagine the same strength magnetic field could produce different profiles for the power spectrum. However the way my professor puts it he makes it seem like he wants me to look and see "oh our 60 hz signal is at 1x10^-4, so that corresponds to a field of x Tesla"

Well, I guess I don't have any more questions but if anyone could explain to me (or show me) how to go about calculating the strength of the magnetic field from the power spectrum that would be incredible.

• How is the magnetic field modulated? Is $B(t)$ periodic in time? – Mark H Apr 12 '17 at 22:24
• Oh yes we have a function generator which creates an AC current (which I have set at ~100 hz, in order to separate from any common noise (15 hz and the higher harmonics). My professor said you can represent the magnetic field as $$B(t) = Asin(\omega t)$$ with $\omega$ set to 100 hz and then A representing the Amplitude. He implies I should be able to find A from the amplitudes in the power spectrum. – Kyle O'Connell Apr 13 '17 at 1:42
• It would bee nice if you really knew the flux density at the given location. You could create a correlation matrix which shows the dependencies between the spectrum and the flux density. It would be just some simple matrix operations, which should be implemented in whatever programm you want to use. A 2D histogram plot with the correlation matrix shows the depencies between field and frequencies nicely. – WalyKu Apr 13 '17 at 11:31

Relating the power in a frequency spectrum can be both straightforward and not depending on how the signal varies with the magnetic field. Multiplying the original signal by a constant scales the Fourier transform by an equal amount, so doubling the magnetic field strength should double the amplitude of every frequency. However, that assumes you are directly measuring the field.

I'm going to make some assumptions about your setup and show a possible analysis.

1. The laser power is constant.
2. The two polarizers (one magnetically modulated and one constant) are crossed so that zero magnetic field means zero light is transmitted.
3. The rotation of the polarizing angle in the magnetically modulated polarizer is proportional to the magnetic field strength.
4. The laser shines through both polarizers before hitting the laser power meter.

The amplitude of light let through the polarizers when the angle between them is $\theta$ off perpendicular is $$A = a\sin(\theta).$$ The measuring device actually detects power, so the signal is $$P = a^2\sin^2(\theta),$$ where $a^2$ is the total input laser power. The angle $\theta(t)$ is given by $kB(t),$ where $k$ is some constant empirically derived from the polarizer material and dimensions and $B(t) = B_0\sin(\omega t)$ is the magnetic field strength parallel to the light. As a function of time, the instantaneous power is $$P = a^2\sin^2(kB_0\sin(\omega t)).$$ Assuming the angle change $\theta$ is small, we can say that $\sin\theta \approx \theta.$ $$P = a^2(kB_0\sin(\omega t))^2 = a^2k^2B_0^2\sin^2(\omega t).$$ From the double-angle trig formulas, we know that a $\sin^2$ signal is really a sinusoid at twice the original frequency with a DC offset. So, the laser power transmitted varies at twice the frequency of the magnet. The amplitude of any frequency of the oscillation will be proportional to the square of the maximum magnetic field.

Getting the magnetic field from this can be complicated since you need to know $k$ from the varying polarizer, the total laser power, and how that laser power varies over time. You also need to check how the FFT of your software deals with multiplicative constants ($1/2\pi$, $1$, $1/\sqrt{2\pi}$, etc.).

If everything was ideal, you would measure the amplitude of the power oscillations at $2\omega,$ since every other frequency would be from noise, imperfections in the function generator waveform, and other uninteresting sources.