Extending geometrical optics: A Lagrangian theory for vector waves

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.

Parametric decay of plasma waves near the upper-hybrid resonance

“…This can be identified as a manifestation of polarization ("spin") effects that were recently discussed for both classical and quantum waves in Refs. [21,22].…”

Quasioptical modeling of wave beams with and without mode conversion. I. Basic theory

“…If D H2 is neglected, the envelope equation (119) becomes a Dirac-type equation similar to those considered in the context of XGO [24][25][26][27][28] and also, for instance, in Refs. [58,59].…”

“…I), except here it is generalized to an arbitrary metric. This term causes polarization-driven bending of the ray trajectories, which is missed in traditional GO; also, it causes mode conversion, if more than one active mode is present [24][25][26][27][28]. These effects are is discussed in further detail in Sec.…”

Variational principles for dissipative (sub)systems, with applications to the theory of linear dispersion and geometrical optics

“…This implies that conservation laws are also generally preserved, so a model inherits key features of the original system notwithstanding the reduction [1]. One of the areas that particularly benefit from this fact is wave theory [2,3], where even crude approximations to an action functional often yield insightful reduced models [4][5][6][7][8][9][10][11].…”