Optical pulse propagation with minimal approximations

Propagation equations for optical pulses are needed to assist in describing applications in ever more extreme situations—including those in metamaterials with linear and nonlinear magnetic responses. Here I show how to derive a single first-order propagation equation using a minimum of approximations and a straightforward “factorization” mathematical scheme. The approach generates exact coupled bidirectional equations, after which it is clear that the description can be reduced to a single unidirectional first-order wave equation by means of a simple “slow evolution” approximation, where the optical pulse changes little over the distance of one wavelength. It also allows a direct term-to-term comparison of an exact bidirectional theory with the approximate unidirectional theory.

Figure 1

An ordinary reflection at an interface between media with permittivities ${\epsilon }_1$ and ${\epsilon }_2$, in a $t$-propagated picture. An incoming pulse propagates forward (in $t$) and evolves forward (in $z$) until it reaches an interface, whereupon it splits into a transmitted pulse and a normal reflected pulse; the reflected pulse then evolves backward in space as both transmitted and reflected pulses continue to propagate forward in time.

Figure 2

A reflection at an interface between media with permittivities ${\epsilon }_1$ and ${\epsilon }_2$, in a $z$-propagated picture. An incoming pulse propagates forward (in $z$) and evolves forward (in $t$) until it reaches an interface, whereupon it splits into a transmitted pulse and its reverse reflection; the reverse reflected pulse then evolves backward in time as both transmitted and reflected pulses continue to propagate forward in space. A reverse reflection is the means by which a spatially propagated system represents a pulse propagating backward in $z$ so it cancels the ordinary reflection missing from the initial conditions.