The Ray Equations

This chapter continues the variational focus by first deriving the Euler–Lagrange equation for the optical path length functional with effective refractive index from previous chapters. Next, we show under what conditions one can find a minimizer of this functional in a particular class of allowable trajectories. This is then extended to the case of rectifiable curves when the director is at least \(C^1\) smooth. This uses a lower semicontinuity argument and the Dunford–Pettis Theorem. We then consider the Hamiltonian formulation (rather than the usual Lagrangian formulation) and provide a brief review of Riemannian geometry concepts necessary to understand light rays as geodesics under a certain geometry, i.e., with a certain metric. We obtain the geodesic equations for the optical path length functional and provide a number of examples. Finally, we discuss broken extremals, which are typical in optics problems when an abrupt change of materials occurs. We discuss the possibility of obtaining discontinuous extremals, with explicit calculations done in two dimensions.