Consider an optical experiment with photons or light pulses.

Is there an optical element that acts in the polarisation degree of freedom like the unitary $$ U = \frac 1 {\sqrt 2} \begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix}\quad \text ? $$ I choosed the basis such that $$ |H\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}\;,\quad |V\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix} $$

If yes, then a photon that passes this device twice would expirience a flip in the polarisation: $$ U^2 = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\;, $$ like $U|H\rangle = |V\rangle$ and $U|V\rangle = -|H\rangle$.

Does it exist?

  • $\begingroup$ The result of passing through this element twice would be to "flip the polarization" - or if you like, it would be a 180 degree phase shift. That would just be a $\lambda/2$ element. But something that does this "after you pass through twice" is more interesting. What you are asking for is an element that rotates the polarization angle by 90 degrees. I have a hunch that you could achieve this with a birefringent element, but I'm not sure that can be done when you don't know the incident polarization angle. $\endgroup$
    – Floris
    Commented Oct 16, 2014 at 14:25

1 Answer 1


This is a typical matrix for an optically active element that rotates the light polarisation. A cuvette of water with sugar will do the job. Proportional to the sugar concentration, you can obtain arbitrary wave rotation.

Note that the U matrix has imaginary eigenvectors (1+i)/sqrt(2) and (1-i)/sqrt(2). Accordingly, unlike λ/4 plates, the eigenwaves of such a system are left- and right-circularly polarized.

  • $\begingroup$ Note also that optical activity is time reversible. A different result can be obtained by defining both off-diagonal elements of U as imaginary, in which case the forward and backward waves would be rotated in the same direction breaking the time-reversal symmetry. This effect occurs e.g. in statically magnetized bismuth-iron-grenade and is employed in optical insulators. $\endgroup$
    – dominecf
    Commented Aug 18, 2015 at 20:11

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