Adding depolarizing decomposition

[Jashcraf]

Hello! I was interested in learning Julia after some persistent persuasion and decided to add a little bit to this package as a first pass because I too exist in Mueller calculus.

Specifically, I’m trying to add the depolarizing decomposition of a Mueller matrix so that you can analytically separate out the Depolarizer, Retarder, and Diattenuator from a single Mueller matrix.

For the time being, here’s the Diattenuator function

using LinearAlgebra

function decompose_diattenuator(M)

    T = M[1, 1]
    diattenuation_vector = M[1, 2:4] / T

    D = sqrt(dot(diattenuation_vector, diattenuation_vector))
    mD = sqrt(1 - D^2)
    diattenuation_norm = diattenuation_vector # / mD
    DD = kron(diattenuation_vector, transpose(diattenuation_vector))

    # Create the diattenuator
    I = diagm([1., 1., 1.])
    inner_diattenuator = (I * mD) + (1 - mD) * DD

    # Put it all together
    Md = zeros(4, 4)
    Md[1, 1] = 1.
    Md[1, 2:4] = diattenuation_vector
    Md[2:4, 1] = diattenuation_vector
    Md[2:4, 2:4] = inner_diattenuator
    Md = Md * T

    return Md
end

Does this sound like it would be a useful inclusion to Mueller.jl? If so, I can get started on a PR

[Jashcraf]

For reference, I have a similar set of functions implemented in Python in my Katsu package
https://github.com/Jashcraf/katsu/blob/deea56fdcba0f60a4c65ef42a9b03f26d70950bb/katsu/mueller.py#L489