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Radar is "just" microwave-frequency electromagnetic radiation. The radar antenna emits a wave and then measures the return wave to "see" objects. This return wave can come from other planes, terrain features, and depending on radar design, rainclouds. In order to "see" through the nose of the plane, these electromagnetic waves must pass through the nose without producing any signal back towards the antenna or suffering much energy loss.

The nose covering over a radar is called the radome. Usually the radome is made of something transparent to the wavelengths used by the radar (aka microwave radiation). This is usually a lightweight plastic or foam. The key property is that the material has what we call a low permittivity. You can think of this as the material not interacting strongly with electromagnetic radiation.

>microwave-frequency

Radio frequency. It's in the name. RAdio Direction and Ranging.

Edit: I stand corrected. I looked it up, and although it may have started with just radio frequencies, it does indeed, now, use multiple frequency ranges, despite still being called radar (I also acknowledge I got the acronym wrong: it is indeed radio direction and ranging)

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Radars operate both in the microwave and radio frequency domains of the EM spectrum

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Most aircraft radars that I am aware of are somewhere in the X to Ka band, about 7-40GHz, which is indeed right in the middle of the microwave range as commonly defined.

So if we could see in the frequencies that radar sees, we would see the radome as a transparent bubble? Neat!

Well yes, and many other things like most interior walls and basically all relatively thin non-metallic and non-watery things which let your cellular or wifi signal pass through just fine with just some attenuation. Resolution’s limited by wavelength though, the “pixels” of microwave vision are on the order of 1..10 cm or so at best. And you’d need big “eyes”.