Faraday rotation

Last Updated on June 5, 2026 by John Berry

Most of the work on this site is based on empirical work by engineers and scientists. Most of it allows radio amateurs to treat the plasma of the atmosphere as a black box. We describe it without opening the box and having to plough through the work of 19th century scientists. Most of it, that is, until we hit Faraday rotation during EME moonbounce. Here we need to understand rotation as an observable effect because it may result in a decibel loss in received signal.

Over the past year or so, I’ve experienced this black box approach in the chat rooms of EME. For much of the time, the path from the Earth to the Moon and back can be modelled with a core path budget that I’ve outlined in a calculator on the path budget page. Sure, there’s sky noise degradation, but radio amateurs just avoid the time when the Moon is in a bad place. Of course, there’s libration fading, but Joe Taylor has taken care of that in WSJT-X. And from time to time, those that can’t elevate their antennas exploit ground gain as the Moon appears over the horizon. But when a received signal is just not there when it should be, Faraday gets blamed.

Perhaps that’s correct. Something as complicated as Faraday rotation deserves to be blamed for the inability to QSO. But it does deserve a bit more of an explanation than just saying, ‘Mr Faraday is at play’.

To the Moon and back

We need to start with a core ray diagram for EME signals. One station points antennas at the Moon and transmits a signal through the troposphere to the ionosphere. The signal transits the ionosphere, reflects or scatters off the Moon, returning to the top of the ionosphere. It transits the ionosphere again, back into the troposphere, and back to the receiving station. 

The key thing is that there are two transits of the ionosphere. And it’s in the ionosphere that Faraday rotation occurs. As the above diagram also shows, it’s there that the wave is refracted twice.

The ionosphere has some important characteristics. As a plasma, it is a soup of ions and free electrons. The pair exist in equilibrium but periodically they recombine and new pairs split apart to yield the all-important electrons. 

There are three important frequencies associated with the ionosphere: f, the radio or signal frequency, fg, the gyro frequency, and fp, the plasma frequency. 

As a plasma, the ionosphere has a refractive index, μ. This is important because Snell’s law is obeyed at all states of transit. The change in wave vector direction k, is given by Snell’s equation:

μ sin φ = μ₀ sin φ₀

Where the left side of the equation describes the incident wave and the right describes the wave upon entering a layer of different refractive index. It also sets up the condition for reflection discussed in other pages on this site.

Under the influence of a field

We start with the notion of a steady state ionosphere. The ionosphere is assumed to be electrically neutral with equal ions and electrons, sitting in a uniform external (Earth’s) magnetic field, B₀. It’s considered a cold plasma, so movement of particles is ignored. 

The notion of a stable ionosphere is a bit of a stretch of course, which is why Faraday rotation is not present for much of the time.

The diagram above shows the vector k representing a plane electromagnetic wave, and the Earth’s magnetic field, B_0. k is travelling along the x axis in the diagram, and the magnetic field is in direction 𝞱. This diagram sets the scene for the field to influence the electrons that propagate the wave and for the vector k to rotate.

Significance of electrons

Consider a wave travelling vertically from Earth that collides with ionosphere electrons. Critically, if the plasma frequency fp equals the wave (or signal) frequency f, a vertical travelling wave will be reflected from the ionosphere. If the signal frequency exceeds the plasma frequency, the wave will penetrate the layer. This gives the notion of a critical incidence frequency discussed in other pages. It also gives the notion that HF frequencies get reflected while VHF and above penetrate. Of course, it’s not as simple as that but it’s enough for now. Faraday reflection occurs above the critical frequency, typically in the upper HF and VHF frequencies.

Now here’s the important bit.

The Appleton-Hartree equation describing the detail of propagation in the ionosphere shows that the refractive index of the ionosphere is in fact bivariate.  As a result, the wave splits in two – into the ordinary or o wave, and the extraordinary wave or x wave. Together they are known as characteristic waves. The two waves react differently to the magnetic field and rotate in different directions. The two then transit the ionosphere independently.

Behaving differently

The o wave and the x waves behave differently depending on the angle between the wave vector and the magnetic field. We saw one result of this above illustrating one limit condition that realised a critical angle.

The following diagram describes the scenario after the wave has travelled a distance in the ionosphere. This I think describes Faraday rotation well.

The angle of resultant rotation Ω is given by the equation:

Where the integral is over the ionosphere path length s.

This means that the degree of rotation depends on the inverse of the signal frequency squared, the electron density in the ionosphere at that place and time, the strength of the Earth’s magnetic field at that place and time, and the angle of incidence of the wave. The frequency term means that Faraday rotation is significant at VHF, but quickly diminishes through UHF and SHF. The theta 𝞱 term means that Faraday rotation is greatest for slant paths where cos 𝞱 tends to 1. There is also the term s, describing the distance over which the wave travels in the ionosphere and hence the time for which it is influenced by the field. And finally, rotation varies as the electron density which is maximum at sunspot maximum, varies diurnally, and is significantly influenced over a number of time periods by anomalous effects.

Recombination

On exiting the ionosphere the o and x waves recombine to be linearly polarised with amplitude A at angle Ω. Whether that shift in polarisation matters now depends on what happens over the reverse path, and upon the antenna capabilities of the receiving station.

The example here of a wave parallel to the field is a single condition to illustrate the point. The wave and field may be at all manner of angles.

Return path

There is no reason why the return path should be any different and hence why it might use any different theory. But of course, the signal vector k is now in the opposite direction, and the segment of ionosphere through which the wave passes is different for bistatic foreign QSOs. Metaphorically if not exactly, it was over the Atlantic, and now it’s over the Pacific. And both electron density and Earth’s magnetic field vary considerably across the globe. They were one level, and now they’re another. To cap it all, 𝞱 is different too. So the o and x waves will be different, and behave differently, and so there is distinct non-reciprocity. 

For own-echo monitoring, the wave transits the exact same ionospheric column, there and back.

The vector diagram above used to model the resultant signal vector and rotation angle Ω, must be redrawn in reverse and the equation for Ω must be recalculated. 

It is possible therefore that the Faraday rotation may occur predominately in the uplink, or predominantly in the downlink. It’s also possible that rotation occurs about equally in both uplink and downlink and creates significant polarisation distortion at the receiver. Practical values of Ω seem to suggest angles of up to a few tens of degrees during each transit.

Here’s a graphical summary.

Conclusion

It’s all very irregular – many unmeasurable variables varying in unknown ways. And we do need to consider that ray trajectories as the waves transit the ionosphere are curved as a result of refraction in the ionosphere. It all this makes real assessment of the degree of rotation impossible to determine in all but a research environment.  What we can say with certainty is that Faraday rotation is not always present. And when it is, it varies with frequency – worst on 144MHz, moderate on 432MHz and least on 1296MHz and above.

This final diagram attempts to summarise the scenario. It shows a downlink path. The uplink path has already been reflected or scattered from the Moon and is now heading for Earth. The resulting received signal is the sum of all arrivals of varying amplitude, phase, and polarisation. We’d have to resort to statistics to describe this.

The ionosphere is also not an homogenous medium. It is full of irregularities with differing electron densities. Layers form and disperse. Some layers become metal-like and reflect. And in any case, rays from stations on Earth and those returning from the Moon are not singular. There are pluralities of rays, each acted on differently. Their effect at the receiver is the aggregation of many direct rays, and of many multi-path rays in the receiver environment.

Received signal

The arrivals presented to the receiver will have various amplitudes, phases, Doppler frequencies, and polarisations. Critically, if the main arrivals no longer have the same polarisation as the wave launched in the uplink (and that expected at the receiver), they will experience polarisation discrimination which manifests as a decibel loss in aggregate received signal. Perhaps it might be useful to flip the receiver antennas to to be vertical polarised; or maybe not.

So it’s no surprise that many radio amateurs treat it all as a black box and blame Faraday without further consideration. To do otherwise is near impossible.