What's the scattering matrix for a PBS (polarization beam splitter)? Is it just unitary? If one polarization never couples into another polarization (then there's a lot of zeroes in that 4x4 matrix) - is that impossible somehow?
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I think it's simplest just to propagate S and P separately, Then use the ABCD matrix for a mirror for the reflected polarization and the matrix for transmission (or no matrix at all :-) ) for the transmitted polarization.
A couple possibly useful references here.
One of a series of powerpoint lectures -- I haven't looked at the contents of the other 37.
Wikipedia's pages for Jones calculus and ABCD ray tracing
Carl Witthoft
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1He asked about the S-matrix in the quantum sense, not Jones or Mueller formalisms (which are purely classical). – daaxix Aug 09 '14 at 15:06
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@daaxix I don't think that's right: one can reinterpret Jones matrices as unitary operators in $SU(2)$ acting on pure polarization states, the Jones vector being the superposition weights. The Jones matrices are the on-diagonal blocks of the S-matrix. Only when one must model reflexions are these diagonal blocks nonunitary (otherwise they are an excellent approximation). One one can think of a Mueller matrix as containing the image under the (big A, i.e. Lie group rather than algebra) adjoint representation $\mathrm{Ad}$ of $SU(2)$. The Mueller matrix acts on the image of a density matrix ... – Selene Routley Aug 12 '15 at 06:27
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@daaxix ... for the forward propagating part of the light under $\mathrm{Ad}$, thus it generalizes the Jones matrix to mixed forward propagating states. See here for further details. – Selene Routley Aug 12 '15 at 06:27
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quantum theory of light
Yes, there exists a quantum theory of light which turns into an operator form Maxwell's equations and gives a wave function to a photon. The classical electromagnetic wave emerges from a confluence of individual photons with energy E=h*nu , the frequency of the classical wave. It is not a simple formulation, as one can see in this blog entry here. .
Once one deals with beams and polarizations and beam splitters it is a lot simpler/elegant to use the classical formulations as given in the answer by Carl, since there exists a rigorous correspondence in the mathematical description between the quantum states and the classical ones for the case of light.
anna v
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Do you have any references for the rigorous correspondence, I work on the classical side and haven't seen it derived how the transverse wave coherence properties and individual photon polarization correspond, what happens to the phase uncertainty as the number of photons goes to 1? (I'm not talking about Fock states here building up a wave, but going from Jones matrices/Stokes vectors to polarization of single photons), I ask because classical polarization is relationship of phase. Also, the blog linked had so many ads it locked up my browser... – daaxix Aug 16 '15 at 20:58
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Try another browser or PC, because it is worth reading that blog. The simplest is this https://en.wikipedia.org/wiki/Spin_angular_momentum_of_light . Note that the photon has spin +/-1 to its direction of motion and builds up a classical wave with polarisation perpendicular to its direction. – anna v Aug 17 '15 at 03:36
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anna v, that wikipedia article doesn't add any insight into the "building up the wave" part, nor that there is a correspondence or the derivation of that correspondence. Just because the math is similar doesn't mean it is the same thing...the "spin angular momentum" of circularly polarized light can be derived and even experimentally measured completely within a classical framework...it is true that descriptions of polarized light, which are basically projection operators onto analyzer states, are similar to QM things, but I would like a good peer reviewed reference or Arxiv link... – daaxix Aug 17 '15 at 06:02
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why don't you ask this in http://www.physicsoverflow.org/questions/main . try the minimal link for Motl's blog entry http://motls.blogspot.com/2011/11/how-classical-fields-particles-emerge.html?m=1 – anna v Aug 17 '15 at 06:57
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