Light Polarization and Jones Matrices

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Light reflecting off a mirror actually penetrates a short distance into the
mirror surface material. In metals, this distance is very short (much less
than a wavelength) and so can be neglected. But metals tend to also absorb
~10% of the light, which is undesirable. Today’s modern multilayer
dielectric coatings, on the other hand, don’t absorb any light, but
light tends to penetrate further into them, usually further for the ppolarization
than for the s-polarization. Suppose that light at a wavelength
of 1um with 45 degree linear polarization reflects off a dielectric-coated
mirror (let n = 2 for both polarizations in the dielectric coating). If the ppolarization
penetrates 2um into the dielectric material, and the spolarization
only penetrates 1.875um, what will be the polarization of
the reflected light? What if the light has a wavelength of 500nm (use the
same refractive indices and penetration depths)? Write down Jones matrices
for the mirror for the two cases.

I know (sorry for poor formatting)
JQ=1√2⋅(1+icos(2θ) isin(2θ)
isin(2θ) 1−icos(2θ))
but I'm not sure how to use the wavelength or n to solve this - or even how it relates to the different polarizations (aside from the ones in the chart at https://en.wikipedia.org/wiki/Jones_calculus).

Any help would be much appreciated!

Thank you.

 

[Daz]

The question gives you a very simplistic model of a dielectric mirror: assume that the p-polarised component travels 2um into the surface before it “reflects” and starts coming back out. i.e. a round-trip of 4um. The s-polarised component, on the other hand doesn’t have to travel so far. So the p- component has to go further and therefore after reflection it will be lagging in phase.

What you need to do is work out how much that phase lag will be and then express it as a Jones vector.

Remember that the phase needs to be expressed as an angle where a full cycle is 360degs (or 2PI radians). A full cycle corresponds to one wavelength travelled, so if you can find the path difference between the s- and p-components, then you’re nearly there. Work out what fraction of a wavelength that corresponds to and remember that the wavelength in the mirror is different because of its refractive index.

If n=2 what will the wavelength be inside the mirror?

P.S. You're supposed to use the homework template and show your own attempt at a solution when asking for help.