Abstract
To model electromagnetic wave propagation for coherent communications without polarization dependent
losses, the unitary
$2 \times 2$
Jones transfer matrix formalism is
typically used. In this study, we propose an alternative formalism to describe such transformations based on rotations
in four-dimensional (4d) Euclidean space. This formalism is usually more attractive from a communication theoretical
perspective, since decisions and symbol errors can be related to geometric concepts such as Euclidean distances
between points and decision boundaries. Since 4d rotations is a richer description than the conventional Jones
calculus, having six rather than four degrees of freedom (DOF), we propose an extension of the Jones calculus to
handle all six DOF. In addition, we show that the two extra DOF in the 4d description represents transformations that
are nonphysical for propagating photons, since they does not obey the fundamental quantum mechanical boson commutation
relations. Finally, we exemplify on how the nonphysical rotations can change the polarization-phase degeneracy of
well-known constellations such as single-polarization QPSK, polarization-multiplexed (PM-)QPSK and
polarization-switched (PS-) QPSK. For example, we show how PM-QPSK, which is well known to consist of four
polarization states each having four-fold phase degeneracy, can be represented as eight states of polarizations, each
with binary phase degeneracy.
© 2014 IEEE
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