Optical Eigenmodes

At its foundation quantum mechanics relies on the definition of eigenstates. These eigenstates correspond to orthogonal wave functions such as the atomic orbital wave functions of the electron in the Hydrogen atom (see figure left, Hydrogen orbitals ). Many of the properties that can be observed in quantum mechanics directly relate to the decomposition of any quantum system onto a set of orthogonal eigenstates. It is the orthogonality of these states that offers a unique way to identify each of the eigenstate by a set of measurable quantities. The typical example is the electron quantum state that is uniquely defined by the principal, orbital, magnetic and spin quantum numbers.

The optical eigenmodes expands this notion to the field of classical optics governed by Maxwell's equations. In this context, the concept of optical eigenmodes is directly related to canonical conserving quantities. Indeed, for each of these quantities, it is possible to define a linear operator which is associated with a set of orthogonal electromagnetic solutions of Maxwell's equations. 

Optical orbital eigenmodes

A colourful illustration are the optical eigenmodes of the angular momentum operator. These result from the optical momentum 

\[ {\bf M}={\bf r}\wedge {\bf F}\]
where 

\[F_i=\frac{\varepsilon_h\varepsilon_0}{2}\text{Re} (\alpha E_j\partial_i E^*_j)\]

is the optical dipole force.