# Mode expansion inside a waveguide

Physics Asked by user135626 on December 6, 2020

In the classic book by Collin (Foundations for Microwave Engineering, 2nd ed.), the author postulates on p. 278 that, given a waveguide structure that stretches along the $$z$$ axis with open ends (see figure attached), and given a current source $$J$$ inside the region captured by the two transverse planes at $$z=z_{1}$$ and $$z=z_{2}$$ in the wvaeguide, then the electromagnetic fields generated to the right ($$E^{+},H^{+}$$) and to the left ($$E^{-},H^{-}$$) may be given as expansions of the orthogonal waveguide natural modes (eigenfunctions) as:

$$boldsymbol{E}^{+}=sum_{n}C^{+}_{n} (boldsymbol{e_{n}}+boldsymbol{e_{zn}})e^{-ibeta_{n}z} ; z>z_{2}$$
$$boldsymbol{H}^{+}=sum_{n}C^{+}_{n} (boldsymbol{h_{n}}+boldsymbol{h_{zn}})e^{-ibeta_{n}z} ; z>z_{2}$$
$$boldsymbol{E}^{-}=sum_{n}C^{-}_{n} (boldsymbol{e_{n}}-boldsymbol{e_{zn}})e^{ibeta_{n}z} ; z
$$boldsymbol{H}^{-}=sum_{n}C^{-}_{n} (-boldsymbol{h_{n}}+boldsymbol{h_{zn}})e^{ibeta_{n}z} ; z

where $$C$$ are amplitude coefficients, $$e_{n},h_{n}$$ are the transverse components of such modes, while the $$e_{zn},h_{zn}$$ are their longitudinal components. Note that the author uses symbol $$j$$ instead of $$i$$ for $$sqrt{-1}$$ in the image attached. Bold symbols are vectors.

I understand the expansion itself in terms of the modes, but I am not sure why did the author insert a minus sign ($$-$$) in front of $$e_{zn}$$ and $$h_{n}$$ in the last two equations? I assume that explaining one would lead to the other (from curl relations).

Could it be because he assumed, for example, that the $$e_{zn}$$ mode is origially defined to be along the $$+z$$-direction, and by looking into the opposite direction for $$E^{-}$$ we then see this reversed? And shouldn’t inverting $$e^{-ibeta_{n}z}$$ to $$e^{ibeta_{n}z}$$ be the only mathematical operation made in moving from one direction to another?

The direction of the longitudinal field component $textbf{e}_{zn}$ is parallel with the $z$ axis. When the wave is moving in the same direction its propagation factor is $e^{-jbeta_n z}$. This is denoted by the $+$ sign as superscript such as $textbf{E}^+$. The reflected wave denoted by $textbf{E}^-$ is moving backwards, i.e., anti-parallel with the $z$ axis therefore its longitudinal component must have the opposite sign to that of the forward wave, hence the term $-textbf{e}_{zn}$ in (4.107c). The sign of the transversal component $textbf{e}_n$ of the E-field is not affected by the reflection along the z-axis.