# Tutorial: TE and TM modes in a cylindrical waveguide

1. (0 pts) Study the derivation of the TE modes in a rectangular waveguide in Jackson’s section 8.4.
2. (4 pts) Work out the TM modes for the same waveguide. [You should end up with formulas that parallel (8.44-8.46).]
3. (2 pts) Which of the TE and TM modes has the lowest cutoff frequency in a rectangular waveguide with $$a>b$$? Is there only one mode with this cutoff frequency?
4. (3 pts) Find the second-lowest, third-lowest, and fourth-lowest cutoff frequencies in a rectangular waveguide with $$a=1.5b$$. How many modes have each of these cutoff frequencies?
5. (6 pts) Calculate the attenuation constant for the mode with the lowest cutoff frequency in a rectangular waveguide with $$a=2b$$, using the method of Jackson’s section 8.5.]
6. (0 pts) Note that the power flow tends to zero near the cutoff, and the attenuation constant found in step #5 diverges. This result is unphysical, because the wavevector should always be finite. Therefore, the method of section 8.5 fails near the cutoff. [A more general method is described in section 8.6, which you should review.]
7. (2 pts) Consider a waveguide with a circular cross-section. Write the eigenvalue equation in cylindrical coordinates.
8. (2 pts) Use variable separation of show that the solutions can be written as $$\psi(\rho,\phi) =R(\rho)Y(\phi)$$.
9. (2 pts) Show that $$Y(\phi)=e^{im\phi}$$ and explain why $$m$$ must be integer.
10. (2 pts) Use the results of steps #7 and #8 to obtain the ordinary differential equation for $$R(\rho)$$. Identify this equation by name.
11. (2 pts) Recall which special functions represent the solutions of the resulting equation. Identify those that are singular at the origin and explain why they should be discarded.
12. (3 pts) Using the results of steps #9 and #10, establish the procedure for finding the cutoff frequencies of all TE and TM modes in a circular waveguide. Introduce the notation $$\mathrm{TE}_{(m,n)}$$and $$\mathrm{TM}_{(m,n)}$$, where $$m$$ is the integer number from step #8 and $$n$$ is the sequential number of the solution.
13. (4 pts) Find 5 lowest cutoff frequencies and label the corresponding modes using the notation from step #11. Note the degeneracies.
14. (2 pts) In step #13 you should have found that some cutoff frequencies are doubly degenerate. Identify the relation between the relevant special functions that is responsible for this degeneracy.
15. (0 pts) Consider a waveguide whose cross-section is an isosceles right triangle. Note that the variables can not be separated in this case.
16. (1 pts) Note that the isosceles right triangle can be obtained by slicing a square along its diagonal. Using the fact that the mirror reflection along the diagonal is a symmetry operation for the square, argue that the TE and TM modes for the square can be chosen to be even or odd under this reflection.
17. (2 pts) The solutions for the TE and TM modes of a rectangular waveguide were found in section 8.4 and in step #2. Specialize these solutions to the case of a square cross-section and list the $$H_z$$ field configurations for all TE modes and the $$E_z$$ field configurations for all TM modes.
18. (2 pts) Explain why degenerate modes are only defined up to an arbitrary linear combination.
19. (3 pts) Form linear combinations of the modes found in step #17 that relabel the TE and TM modes in terms of their parity (even or odd) under mirror reflection along one the square diagonal where $$x=y$$. Which combinations of labels are possible?
20. (3 pts) Use the modes from step #19 to find the TE and TM modes for the triangular waveguide.
21. (2 pts) Identify the TE and TM modes with the lowest cutoff frequency for the triangular waveguide.

Total: 47 points.