## Complex form of Poynting relationship

Posted by peeterjoot on August 2, 2012

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This is a problem from [1], something that I’d tried back when reading [2] 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

## Answer

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.

# References

[1] G.R. Fowles. *Introduction to modern optics*. Dover Pubns, 1989.

[2] JD Jackson. *Classical Electrodynamics Wiley*. John Wiley and Sons, 2nd edition, 1975.

## daaxix said

Note that this is only valid for a single plane wave, if the field consists of more than a single plane wave there will be cross terms which are not accounted for in your derivationâ€¦

## peeterjoot said

It looks to me that this allows for elliptical or circular polarization too, which are superposition states (1.0.6/7 1.0.7/8 allows for that I think). Do you mean that this only works for waves of the same frequency? If so, I’d agree.