First-principles variational formulation of polarization effects in geometrical optics

The propagation of electromagnetic waves in isotropic dielectric media with local dispersion is studied under the assumption of small but nonvanishing $\lambda /\ell$, where $\lambda$ is the wavelength and $\ell$ is the characteristic inhomogeneity scale. It is commonly known that, due to nonzero $\lambda /\ell$, such waves can experience polarization-driven bending of ray trajectories and polarization dynamics that can be interpreted as the precession of the wave “spin”. The present work reports how Lagrangians describing these effects can be deduced, rather than guessed, within a strictly classical theory. In addition to the commonly known ray Lagrangian that features the Berry connection, a simple alternative Lagrangian is proposed that naturally has a canonical form. The presented theory captures not only the eigenray dynamics but also the dynamics of continuous-wave fields and rays with mixed polarization, or “entangled” waves. The calculation assumes stationary lossless media with isotropic local dispersion, but generalizations to other media are straightforward.

Figure 1

Comparison between ray trajectories predicted by the canonical Lagrangian (77) and the noncanonical Lagrangian (82). The red line represents $\mathrm{X}\left(t\right)-\mathrm{A}(\mathrm{P}\left(t\right))$, where $\mathrm{X}\left(t\right)$ and $\mathrm{P}\left(t\right)$ are found by numerical integration of Eqs. (79) and (80). The blue line shows $\mathrm{x}\left(t\right)$, as found by numerical integration of Eqs. (91) and (92). The black dashed line plots the “spinless” ray trajectory governed by the lowest-order GO ray Lagrangian $L_0$ given by Eq. (94). The refraction index is chosen to be $n\left(\mathrm{x}\right)=1+exp[-(y^2+z^2)/{\ell }^2]$, the characteristic width is $\ell =10$, and $\mathrm{A}$ is chosen in the form (51). The length unit is $a\doteq \sqrt{2}/p_0$. The initial location is ${\mathrm{x}}_0=(0,10,0)$, and the initial momentum is ${\mathrm{p}}_0=(1,0,1)/a$; hence $\epsilon \sim 1/\left(p_0\ell \right)\sim 0.07$. It is seen that the canonical Lagrangian and the noncanonical Lagrangian predict results that, for the specified parameters, are essentially indistinguishable from each other yet differ noticeably from those yielded by $L_0$.