TE Modes of Cylindrical WaveguideEditors:Advisor:Fig.1 shows a cylindrical waveguide which is simply a hollow tube with circular cross section of radiusaandextending along the-direction. There are two sets of modes, TE and TM modes, which can propagate in a cylindrical waveguide. For TE waves in a cylindrical waveguide,and, all other field components can be expressed in terms of.

Fig.1 Configuration of a cylindrical waveguideThe Maxwells equations can be expanded in the cylindrical coordinate asTheandcomponents can be expressed in terms ofandas

For TE waves,, andsatisfies the equation(1)orwhere.To solve this equation, we make use of the separation-of-variables technique, namely,, and substitute into (1) to have

The general solution has the form

Thus,The boundary condition at=arequires that the tangential component ofbe zero, namely,orwhere.Notethat. The first few roots ofare listed in the following Table.

01234

13.831.843.054.205.32

27.025.336.718.029.28

310.178.549.9711.3512.68

The allowed values ofare expressedas, and the associated modes are designated asTEn,mmodes. Explicitly

Field distribution of TE01at transversal cross section(f=2fc)

Field distribution of TE01at longitudinal cross section(f=2fc)

Field distribution of TE01at transversal cross section(f=0.9fc)

Field distribution of TE01at longitudinal cross section(f=0.9fc)

Field distribution of TE11at transversal cross section(f=2fc)

Field distribution of TE11at longitudinal cross section(f=2fc)

Field distribution of TE11at transversal cross section(f=0.9fc)

Field distribution of TE11at longitudinal cross section(f=0.9fc)

Field distribution of TE21at transversal cross section(f=2fc)

Field distribution of TE21at longitudinal cross section(f=2fc)

Field distribution of TE21at transversal cross section(f=0.9fc)

Field distribution of TE21at longitudinal cross section(f=0.9fc)