Figure 6 – A single permanent magnet pole applied behind a ferrofluid cell. Clearly, the light only scatters back to the camera in Figure #6 when the local electric or magnetic dipole resonance within the cell matches the polarization and poynting vector of the incoming light. Since the same photons are driving the resonance within the cell, it is not surprising that the resonances can propagate within the cell as a chain reaction.
But why do the lines of resonance follow the potential energy contours of the applied field? In Figure #5, we see the period of pendulum is a function of the potential 'g'. 'g' is the attraction between the pendulum and the earth. Anyone who has taken Quantum Mechanics knows that frequency of a harmonic oscillator is a function of its potential.
The ferrofluid cells are a driven system with a large applied magnetic field, first the incoming photons cause the system to oscillate and then the oscillations cause the system to scatter the same photons that caused it to oscillate.
This wave of resonance is free to propagate to other oscillators of the same frequency. But where in the ferrofluid cell do you find other oscillators of the same frequency? From the RLC model, remember the ferrimagnetic nanoparticles have magnetoresistance. The oscillation frequency will a function of the magnetoresistance and magnetoresistance is a function of the applied field. Or on the other hand, you can just say that the harmonic oscillator frequencies is a direct function of potential of the applied magnetic field.
Either way, we can now imagine a system of concentric circles of harmonic oscillators with slightly different oscillator frequencies per circle. The driven resonance wave coming from the light source can only effectively propagate to the other resonators like itself, which happens to be along the potential energy contours.
I believe that the reason that the potential energy contours show up in the ferrofluid cell photographs is because that is the path of maximum resonant energy transfer from oscillator to oscillator within the cell. It is the same reason that humans walk along ridge lines instead of rolling hills, it is much more energy efficient to propagate at the same potential.
Figure 7 – Five permanent magnet poles applied behind a ferrofluid cell. Hopefully the reader will a agree that no matter what we call the ferrofluid cell seen in Figure #7, be it a Kerr photonic crystal or a metamirror or the forerunner of a self assembled optical computer with dynamic circuits along magnetic field lines; that the image will stay the same.
Main Body: I have been asked to apply some math to the ferrofluid cells that I study. For example, people wish to know the thickness of the ferrofluid layer in the cell. For some reason saying three to five drops of ferrofluid is not a satisfactory answer.
The ferrofluid cell seen in Figure #8 that I am going to model is made up of BK7 glass window that is 100mm in diameter and 6mm in thickness. Then there is a layer of a mixture of Fe3O4 and Fe2O3 ferrofluid particles of an average 10nm size suspended in light oil. Below the ferrofluid layer is an aluminum surface mirror that 100mm in diameter and 3mm in thickness. The top window is rated at ¼ wave flatness at 650nm.