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The question Fresnel Transmission Coefficient for Magnetic Field is interesting.

Thinking about it led me to reflect upon what little I know of the history of optics, with refraction by lenses and prisms being expressed in terms of an index of refraction, which at lower frequencies (microwaves) was related to the dielectric polarizability.

Today in optics texts it's usually the electric field amplitude rather than the magnetic field amplitude that's calculated, though we could just as well use either one with the proper conversions.

This led me to wonder Are there real-world examples of refraction due to magnetic permeability?

You can't focus light with an iron lens because it's opaque and possibly wouldn't have much permeability at such a high frequency. You might be able to make a microwave lens out of ferrite or some other low-loss medium, but for the purposes of this question only I won't call microwaves "light".

Question: In the wavelength range of IR (say about 10 microns or shorter) to near-UV (say about 100 nm or longer) are there any practical examples or demonstrations of refraction by the magnetic permeability of a material?

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Short answer is no. In that wavelength range, the effect of the magnetic polarizability is almost always much less than the electric polarizability. So even though the index of refraction is defined as $n=\sqrt{\epsilon\mu}$, it is often just written as $n=\sqrt{\epsilon}$ since $\mu\approx 1$ in most materials. And as you pointed out, ferromagnetic materials are usually metals that cannot transmit light.

However, in the recent decade, there has been major advances in fabrication of metamaterials, which exhibit electromagnetic properties not found in natural materials. Metamaterials can be tailored to have arbitrary permittivity and/or permeability (so long as it does not violate physical laws like the Kramers-Kronig relations). So, in theory, it is possible to construct a material with huge magnetic permeability such that the refraction is largely due to the magnetic properties, and not the electric properties.

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    $\begingroup$ Thank you for your answer! "... almost always much less..." is very different than "No." Are you sure that between 10 um and 100 nm no practical examples exist such that within some wavelength range they are transparent and have a $\mu$ that measurably deviates from unity and a component of refraction due to it occurs? Of course it's not common, but that's why I've asked the question in the first place. Thanks! btw any demonstration of a metamaterial in the infrared (shorter than 10 um) would satisfy "are there any practical examples of refraction" and be an acceptable answer. $\endgroup$ – uhoh Jul 26 at 10:01
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For lower frequencies, we can find examples.

One practical example may be at very low/zero frequency. Magnetic permeable materials distort static (or almost-static) magnetic fields. Technically, we can call this refraction.

Oscillating magnetic fields (radio waves) are routinely used to measure thepermeability of materials; what is measured is the effect on the field by the sample, that again it is basically the index of refraction.

Besides lack of transparency, I believe that, as you wrote, at high frequency the permeability of ferromagnets becomes a mess, because the large magnetic domains are "slow" at reacting to the magnetic field. The technique of oscillating magnetic field as before can be used also to assess the size and properties of the magnetic domains, proving that high-frequency permeability differ from the static one.

I think that a transparent fluid/solid loaded with superparamagnetic nanoparticles can show visible refraction effect, but I can't find the source of this belief and not sure up to which frequency it may go; Infrared seems already too much. That's my candidate metamaterial.

Magnons / spin waves should be the way to go to understand the high frequency limit.

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    $\begingroup$ To "Question: In the wavelength range of IR (say about 10 microns or shorter) to near-UV (say about 100 nm or longer) are there any practical examples or demonstrations of refraction by the magnetic permeability of a material?" I can't tell if you are answering "yes" or "no". $\endgroup$ – uhoh Aug 17 at 12:54
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    $\begingroup$ No, I couldn't find examples at IR/visible ;) . I thought that low-frequency examples can help frame the problem. Max resonance frequency I found for superparamagnets is 10 GHZ , 3cm wavelenght, far away from infrared. $\endgroup$ – patta Aug 17 at 12:59
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    $\begingroup$ Something found: arxiv.org/ftp/physics/papers/0701/0701020.pdf $\endgroup$ – patta Aug 20 at 13:06
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    $\begingroup$ A nice presentation: nanohub.org/resources/4275/download/… $\endgroup$ – patta Aug 20 at 13:10
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    $\begingroup$ This is interesting stuff! I'm not sure if these count as examples of true permeability, but they are at least claimed to be magnetic in nature. Nice! $\endgroup$ – uhoh Aug 20 at 13:13

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