# TMZ TME modes, clarification

I'm refering here to Taflove's "computational electrodynamcis, 3rd ed."

He says

Let us assume that the structure being modeled extends to infinity in the z-direction with no change in the shape or position in its transverse cross section. If the incident wave is also uniform in the z-direction, then all partial derivatives of the fields with respect to z must equal zero...

I don't quite understand why the wave should be uniform along z? How could you justify this assumption? Wave's have a finite wavelength, even within media which implies that they vary with respect to position.

Any ideas?

BTW: It seems like this paragraph is also cited in another textbook:

https://books.google.de/books?id=uC70r-R_wF8C&pg=PA208&lpg=PA208&dq=tmz+mode&source=bl&ots=HPALrm6rKU&sig=ACfU3U1ghG1rU2p7F6EAFLJr6PISFtRbGw&hl=de&sa=X&ved=2ahUKEwjKlIHClM_gAhWFzKQKHUswCtwQ6AEwEHoECAMQAQ#v=onepage&q=tmz%20mode&f=false

• Just a note: not having a variation along $z$ means that $E_{x,y,z}=f(x,y)$ not that $E_z = 0$. – Anton Menshov Feb 22 '19 at 13:20

## 1 Answer

I posit that this is better asked on physics.SE, but nevertheless I shall give my take on the answer to this question.

Your problem appears to be the understanding of "uniform wave". Here "uniform" relates to, e.g., uniform plane wave and does not say that each point along the z axis will see the same field configuration.

In a translationally invariant (along z) waveguide each xy-plane is equivalent to any other such plane. You can use Helmholtz-Hodge decomposition for the wave equation and what you will obtain using Fourier's method is that you will have a simple plane wave in the z-direction. The transverse modes (i.e., x,y-modes) will be dependent on the wavenumber $$k_z$$ though.