Complex form of Poynting relationship
Posted by peeterjoot on August 2, 2012
This is a problem from , something that I’d tried back when reading  but in a way that involved Geometric Algebra and the covariant representation of the energy momentum tensor. Let’s try this with plain old complex vector algebra instead.
Question: Average Poynting flux for complex 2D fields (problem 2.4)
Given a complex field phasor representation of the form
Here we allow the components of and to be complex. As usual our fields are defined as the real parts of the phasors
Show that the average Poynting vector has the value
While the text works with two dimensional quantities in the plane, I found this problem easier when tackled in three dimensions. Suppose we write the complex phasor components as
and also write , and , then our (real) fields are
and our Poynting vector before averaging (in these units) is
We are tasked with computing the average of cosines
So, our average Poynting vector is
We have only to compare this to the desired expression
This proves the desired result.
 G.R. Fowles. Introduction to modern optics. Dover Pubns, 1989.
 JD Jackson. Classical Electrodynamics Wiley. John Wiley and Sons, 2nd edition, 1975.