A colleague measures the Mueller matrix of a compact optical component,

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Optics & Photonics · Polarisation and anisotropic media

What an Instrument Can See, Ch Polarisation as a statistical object: coherency and Mueller matrices

Problem

\mathbf M_{\mathrm M}=\operatorname{diag}(1,\,0.80,\,0.80,\,-0.80)

, and computes its Cloude matrix from the chapter's definition, arriving at

\mathbf C_{\mathrm M}=\begin{pmatrix}0.45&0&0&0\\0&0.05&0.40&0\\0&0.40&0.05&0\\0&0&0&0.45\end{pmatrix}.

The colleague then argues: "the trace of

\mathbf C_{\mathrm M}

0.45+0.05+0.05+0.45=1.00

, which is positive, and

\mathbf C_{\mathrm M}

is Hermitian, so its eigenvalues are non-negative and

\mathbf M_{\mathrm M}

passes the physical-realisability test." ((a)) Explain what is wrong with this argument, referring to the trace/determinant rule for a Hermitian matrix's eigenvalues from earlier in the chapter. ((b))

\mathbf C_{\mathrm M}

decouples into the two diagonal entries

0.45

(already eigenvalues) and the

2\times2

block

\begin{pmatrix}0.05&0.40\\0.40&0.05\end{pmatrix}

. Use that block's trace and determinant to find its two eigenvalues. ((c)) List all four eigenvalues of

\mathbf C_{\mathrm M}

and state whether

\mathbf M_{\mathrm M}

is physically realisable.

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