On the “quantum theory of paraxial ray optics according to Dietrich Marcuse” and its connection with a new class of interferometers

The “quantum theory of paraxial ray optics according to Dietrich Marcuse” (DM) has no overt relation with the quantum theory of light. Rather, it is a demonstration of a formal similarity between the transition from classical to quantum mechanics of point particles and recovery of wave optics from ray optics. The Hamiltonian formulation of ray optics in the paraxial approximation has an algebraic form similar to that of the dynamics of a non-relativistic point particle. The fundamental differences between the two cases are: (1) in paraxial ray optics, time is replaced with the geometrical coordinate z along the optic axis (2) the problem is two dimensional rather than three. Conjugate momenta in this analogy are related to the ray inclination in a nearly planar wavefront in the (x, y) plane. DM illustrates that wave-optics can be recovered from ray optics by treating the Hamiltonian conjugate momenta as operators in a manner analogous with quantum theory. “Uncertainty principle” in this analogy puts a lower bound related to the wavelength on the product of uncertainties in simultaneous measurement of coordinates and inclinations of rays in the wavefront. This paper takes the analogy further and looks at the possibility of introducing a geometric phase by adiabatically transporting the “quantum state of a ray” around a closed curve enclosing an area. This leads to a new class of interferometers that circumvent the “uncertainty principle of paraxial ray optics”. Three examples are discussed along with one experimental result. Its current relevance to plasma diagnostics is explained.