# Equation for E- and B- fields in TE11 mode of cylindrical waveguide - Literature doesn't match equations

I'm encountering a very unusual consistency when plotting the vector fields in a cylindrical waveguide mode. In a cylindrical waveguide operating in a given mode, the $\hat{e}_r$ and $\hat{e}_\phi$ components of the $\textbf{E}$-field vector (in a cylindrical coordinate system, with the z-axis aligned longitudinally with the direction of propagation) can be written in terms of the $\textbf{B}$-field vector components as follows: $$E_r(r,\phi,z,t) = -\frac{\omega}{k_z}B_\phi$$ $$E_\phi(r,\phi,z,t) = \frac{\omega}{k_z}B_r$$

With $B_r$ and $B_\phi$ given by, with the $'$ denoting differentiation with respect to r, and where n is a zero of the $l$th Bessel function: $$B_r(r,\phi,z=0,t=0) = \frac{inkc^2}{\omega^2-(kc)^2}J_l'(nr)e^{il\phi}$$ $$B_\phi(r,\phi,z=0,t=0) = \frac{-lkc^2}{\omega^2-(kc)^2}\frac{J_l(nr)}{r}e^{il\phi}$$

These expressions can be found in this link. For good measure, I also independently derived these expressions from scratch; they are indeed correct.

Now, to illustrate the problem I am facing, I have written some Python code that plots the $\textbf{E}$ and $\textbf{B}$-field vectors for a given mode. In this post I shall focus my attention on the well known 'dominant' TE11 mode (this mode is also the easiest case to highlight the inconsistency I am encountering). For the TE11 mode, then, my code gives the following plot for the $\textbf{B}$-field (the waveguide region is the shaded circle; ignore the outside regions):

This plot is exactly correct; there are many images available in the literature for the fields of the TE11 mode. For example, see (here, the B-field is given by the horizontal vector field lines, and the E field the vertical 'bowed' lines):

It's clear that the $\textbf{B}$-fields match. Now comes the problem. If I plot the $\textbf{E}$-field components as given by the expressions at the start of this post, I obtain the following plot:

This clearly does not agree with the $\textbf{E}$-field plots found in the literature (yet arguably shares some similarities). However, the code is plotting the $\textbf{E}$-fields correctly, at least according to the equations given at the start of this post (note I've just used these plots to neatly illustrate my problem; this is a physics question and not a programming related question). You get the same result if you manually hand-calculate a few values of the $\textbf{E}$-field and sketch it out.

For example, the $\textbf{E}$-field vector at the point (0.03, -0.02) on the above plot. Plugging in these position values into the expressions given at the start of this post, one obtains (the units are totally arbitrary; all I care about is the direction of the vector): $$E_r=-1198, E_\phi = -940$$

This corresponds to a vector pointing 'inwards and slightly upwards', which matches what is seen on the plot at this point. You can do this for many points; the plot always agrees with what is calculated.

$\textbf{Summary of problem}$: If the expressions at the start of this post are correct, then the (coloured) vector field plots above must be correct and the all of the plots in the literature must be wrong (unlikely). Alternatively, the expressions at the start of this post are wrong, yet calculate the $\textbf{B}$-field perfectly, whilst getting the $\textbf{E}$-field completely wrong. The dilemma is that the equations for cylindrical waveguide modes are very well documented, and so it is unlikely that they too could be wrong.

I think somewhere in your calculations you've lost a minus sign. In your source the two equations for the $\mathbf{H}$-components are

\begin{align} H_r &\propto \mp \, E_\phi \\[3mm] H_\phi &\propto \pm \, E_r ~.\end{align}

I don't know the convention for the signs, but it is clear that one of the equations needs a minus sign (assuming these equations are correct). However in your expression for $\mathbf{E}$ there are no signs.

Try to plot

\begin{align} E_r(r,\phi,z,t) &= \frac{\omega}{k_z}B_\phi \\[3mm] E_\phi(r,\phi,z,t) &= - \frac{\omega}{k_z}B_r ~, \end{align}

and look what happens. I think that will work. I don't have the means to plot it myself right now, but I calculated some points on paper, which matched the field lines stated in the literature.

• Many thanks. I've looked back over my code and there was *indeed * a missing minus sign - all fixed now! – CrossProduct Sep 28 '16 at 4:33
• Okay, glad to hear that! – schlunma Sep 28 '16 at 13:44