Abstract
We study monochromatic, scalar solutions of the Helmholtz and paraxial wave equations (PWEs) from a field-theoretic point of view. We introduce appropriate time-independent Lagrangian densities for which the Euler–Lagrange equations reproduces either Helmholtz and PWEs 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.
Highlights
Light is an electromagnetic phenomenon which can be described by a field theory governed by Maxwell’s equations
We study monochromatic, scalar solutions of the Helmholtz and paraxial wave equations (PWEs) from a field-theoretic point of view
In this first part of our two-part work, we have mainly developed a Lagrangian theory for scalar, monochromatic optical fields
Summary
Light is an electromagnetic phenomenon which can be described by a field theory governed by Maxwell’s equations. In many practical instances, a vector field representation of light appears redundant and a simpler scalar field description results appropriate. When a scalar description is appropriate, according to the characteristics of the phenomenon under investigation, monochromatic light propagating in free space can be described either by a field y (x, z) = y (x, y, z) obeying the Helmholtz wave equation (HWE). For the case of a monochromatic field of frequency ω, Born and Wolf first rewrite the field as the real part of a time-harmonic complex amplitude, that is V (r)(r, t) = Re{U (r, w)e-iwt} They take the time averages of the energy density and the energy flux vector to obtain conservation laws involving only the time-independent complex field U(r, ω). In our nonstandard approach, propagation along the z-axis of a time-independent field obeying either HWE or PWE, is formally described in the same manner the time evolution of a time-dependent field is depicted in the standard Lagrangian formalism
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
