Field theory of monochromatic optical beams. I

TL;DR

This paper develops a field-theoretic framework for monochromatic optical beams, deriving Lagrangian densities, energy-momentum tensors, and a Dirac form of the Helmholtz equation, revealing connections to cosmology and black hole physics.

Contribution

It introduces a novel Lagrangian approach to scalar optical fields, linking wave equations with field theory concepts and exploring their broader physical implications.

Findings

Derived Lagrangian densities for Helmholtz and paraxial equations.

Calculated energy-momentum tensors and continuity equations.

Presented a Dirac form of the Helmholtz wave equation.

Abstract

We study monochromatic, scalar solutions of the Helmholtz and paraxial wave equations from a field-theoretic point of view. We introduce appropriate time-independent Lagrangian densities for which the Euler-Lagrange equations reproduces either Helmholtz and paraxial wave equations with the $z$-coordinate, associated with the main direction of propagation of the fields, playing the same role of time in standard Lagrangian theory. For both Helmholtz and paraxial scalar fields, we calculate the canonical energy-momentum tensor and determine the continuity equations relating "energy" and "momentum" of the fields. Eventually, the reduction of the Helmholtz wave equation to a useful first-order Dirac form, is presented. This work sheds some light on the intriguing and not so acknowledged connections between angular spectrum representation of optical wavefields, cosmological models and physics of black holes.