Extending geometrical optics: A Lagrangian theory for vector waves

TL;DR

This paper develops a Lagrangian extension of geometrical optics to include polarization effects, such as mode conversion and polarization-driven bending, providing a more accurate wave ray description in vector waves.

Contribution

It introduces a first-principle Lagrangian framework for vector wave rays that incorporates polarization dynamics and effective gauge Hamiltonians.

Findings

Describes polarization-driven divergence of circularly polarized waves in plasma.

Reformulates geometrical optics to include polarization as a classical degree of freedom.

Demonstrates the theory with electromagnetic waves in magnetized plasma.

Abstract

Even when neglecting diffraction effects, the well-known equations of geometrical optics (GO) are not entirely accurate. Traditional GO treats wave rays as classical particles, which are completely described by their coordinates and momenta, but vector-wave rays have another degree of freedom, namely, their polarization. The polarization degree of freedom manifests itself as an effective (classical) "wave spin" that can be assigned to rays and can affect the wave dynamics accordingly. A well-known manifestation of polarization dynamics is mode conversion, which is the linear exchange of quanta between different wave modes and can be interpreted as a rotation of the wave spin. Another, less-known polarization effect is the polarization-driven bending of ray trajectories. This work presents an extension and reformulation of GO as a first-principle Lagrangian theory, whose effective-gauge Hamiltonian governs the aforementioned polarization phenomena simultaneously. As an example, the theory is applied to describe the polarization-driven divergence of right-hand and left-hand circularly polarized electromagnetic waves in weakly magnetized plasma.