TE Modes of
Cylindrical Waveguide
Editors:羅牧杰、林晉宇
Advisor:江簡富教授
Fig.1 shows a
cylindrical waveguide which is simply a hollow tube with circular cross section
of radius a and extending 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 waveguide
The Maxwell’s equations can be expanded in the cylindrical coordinate as
The 
and 
components can be expressed in terms of and 
as
For TE waves, 
,
and 
satisfies the equation
… (1)
or
where 
.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 
=
a requires that the tangential component
of 
be zero, namely,
or
where
.
Note that 
.
The first few roots of are listed in the following Table.
|
|
0 |
1 |
2 |
3 |
4 |
|
1 |
3.83 |
1.84 |
3.05 |
4.20 |
5.32 |
|
2 |
7.02 |
5.33 |
6.71 |
8.02 |
9.28 |
|
3 |
10.17 |
8.54 |
9.97 |
11.35 |
12.68 |
The allowed values
of 
are expressed as 
,
and the associated modes are designated as TEn,m
modes. Explicitly
Field distribution of TE01 at transversal cross section ( f =2fc )
Field distribution of TE01 at longitudinal cross section ( f =2fc )
Field distribution of TE01 at transversal cross section ( f =0.9fc )
Field distribution
of TE01 at longitudinal cross section ( f =0.9fc )
Field distribution
of TE11 at transversal cross section ( f =2fc )
Field distribution
of TE11 at longitudinal cross section ( f =2fc )
Field distribution
of TE11 at transversal cross section ( f =0.9fc )
Field distribution
of TE11 at longitudinal cross section ( f =0.9fc )
Field distribution
of TE21 at transversal cross section ( f =2fc )
Field distribution of TE21 at longitudinal cross section ( f =2fc )
Field distribution of TE21 at transversal cross section ( f =0.9fc )
Field distribution of TE21 at longitudinal cross section ( f =0.9fc )