WAVEGUIDES (& CAVITY RESONATORS)
AND
DIELECTRIC WAVEGUIDES (OPTICAL FIBERS)
導波管(&共振腔)與介質導波管(光纖)
When the frequency is at
microwave range
(f >4 GHz ?), the losses of wave
in a
two-conductor transmission line
(due to
imperfect conductor
and
lossy
dielectric
) become large, it is then more efficient to use a
hollow one-conductor
waveguide.
Also for the
high-power transmission
the solid waveguide structure
is preferred.
Transmission lines are
two-conductor lines which basically support
TEM wave
.
Waveguides
are
single-conductor
lines and
can not support TEM wave
(proof ?)
. =>
TE (transverse electric)
or
TM (transverse magnetic) waves
(
discused later
)
Horn antenna (
號角天線
)fed by waveguides
in a reflector antenna system of a
SNG vehicle.
*
1897 :
Lord Rayleigh
mathematically proved that
EM wave propagation in
the waveguide
was possible.
*
1936 :
G. Southworth
of Bell Telephone Laboratories and W. Barrow of M.I.T.
experimentally
demonstrated the EM wave propagation in waveguides.
Waveguides are often bulky & expensive. Recently with
silicon-based
micromaching technology (MEMS)
, hollow
metallic waveguides
have been fabricated
for
operation at optical frequencies
=>
compared with optical fiber ?
* J. N. McMullin, “Hollow metallic waveguides in silicon-V grooves, IEEE Photonics Technology Letters,
5(9), pp. 1080-1082, Sep., 1993.
Klystron
調速管
(
微波源
)
Magnetron
磁控管
(
微波源
)
TWT
行波管
(
微波放大器
)
(traveling wave tube)
One-conductor waveguide
fed by
Two-conductor
transmission line central “probe”
13-2 Waveguide Modes
z
j
t
t
j
z
e
e
y
x
H
H
e
e
y
x
E
E
,
,
0
0
x
y
O
z
Consider an
uniform waveguide
(
constant uniform cross section
along z
direction
), the wave propagating along the uniform waveguiding structure can be
assumed as :
j
e
e
y
x
H
t
z
y
x
H
e
e
y
x
E
t
z
y
x
E
t
j
z
t
j
z
,
,
,
,
,
,
,
,
0
0
Note
: 1. Propagation constant
is unknown at this moment
2.
E
0
&
H
0
are only
functions of (x, y)
, but they have
a
z
components
!
=>
E
x y
a E
x y
a E
x y
a E
x y
H
x y
a H
x y
a H
x y
a H
x y
x
x
y
y
z
z
x
x
y
y
z
z
0
0
0
0
0
0
0
0
,
,
,
,
,
,
,
,
Since
E H
,
fields in a
charge-free region
satisfy homogeneous vector Helmholtz's
equations :
2
2
2
2
0
0
E
k E
H
k H
k
; &
3-D Laplacian operator
can be expressed as
2
2
2
2
2
t
z
t
z
: cross section coordinates
: longitudinal coordinates
=> In
Cartesian coordinates
x y z
, ,
:
2
2
2
2
2
2
y
x
xy
t
0
)
(
0
)
(
0
0
2
2
2
2
2
2
2
2
2
2
2
2
H
k
H
E
k
E
H
k
H
E
k
E
z
xy
z
xy
since
H
y
x
H
e
y
x
H
e
e
y
x
H
H
E
y
x
E
e
y
x
E
e
e
y
x
E
E
z
z
z
z
z
z
z
z
z
z
z
z
2
0
2
0
0
2
0
2
0
0
,
,
,
,
,
,
2
2
2
2
2
2
2
2
2
2
2
2
PDE
order
2nd
)
(let
)
(
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
0
0
0
)
(
0
)
(
0
)
(
0
)
(
2
2
2
2
h
k
H
h
H
E
h
E
H
k
H
E
k
E
H
k
H
E
k
E
xy
xy
xy
xy
z
xy
z
xy
From
E
j
H
From
H
j
E
z
x
y
y
z
x
x
y
z
H
j
y
E
x
E
H
j
x
E
E
H
j
E
y
E
0
0
0
0
0
0
0
0
0
z
x
y
y
z
x
x
y
z
E
j
y
H
x
H
E
j
x
H
H
E
j
H
y
H
0
0
0
0
0
0
0
0
0
It can be proven that
(
E
0
x
,
E
0
y
,
H
0
x
,
H
0
y
)
fields can be expressed by
(
E
0
z
,
H
0
z
)
x
E
j
y
H
h
H
x
H
j
y
E
h
E
y
E
j
x
H
h
H
y
H
j
x
E
h
E
z
z
y
z
z
y
z
z
x
z
z
x
0
0
2
0
0
0
2
0
0
0
2
0
0
0
2
0
1
1
1
1
TE (transverse electric) mode
TM (transverse magnetic) mode
E
z
0
,
H
z
H
0
z
e
z
0
E
z
E
0
z
e
z
0
,
H
z
0
)
(
=>
2
xy
H
0
z
h H
2
0
z
0 (*)
2
xy
E
0
z
h E
2
0
z
0 (**)
Note: 1.
if
E
0
z
and
H
0
z
can be determined,
all other components
can be obtained.
2. Exact solutions depend on
waveguide cross-section geometry
&
boundary
conditions
at the interface of waveguide conductor & inside dielectric .
13-3 Metal Waveguides
13-3-1
TM modes in Rectangular Waveguides
longitudinal components:
a E
z
z
a E
z
0
z
x y e
,
z
0
a H
z
z
0
(**) =>
2
xy
E
0
z
h
2
E
0
z
0
h
2
2
k
2
&
x
E
h
j
x
E
j
y
H
h
H
y
E
h
j
y
E
j
x
H
h
H
y
E
h
x
H
j
y
E
h
E
x
E
h
y
H
j
x
E
h
E
z
z
z
y
z
z
z
x
z
z
z
y
z
z
z
x
0
2
0
0
2
0
0
2
0
0
2
0
0
2
0
0
2
0
0
2
0
0
2
0
1
1
1
1
F
rom boundary conditions of the waveguide
tangential
E
-
field
=
0
:
E
0
z
(
0
,y
)
=E
0
z
(
a,y
)
=E
0
z
(
x,
0
)
=E
0
z
(
x,b
)
=
0
=> determine
E
0
z
x
,
y
&
eigenvalue
h
2
=> then
(
E
0
x
,
E
0
y
,
H
0
x
,
H
0
y
)
)
,
(
E
x
H
x
)
,
(
E
y
H
y
x
z
y
b
a
z
E
z
ˆ
1.
By using separation of variables
E
0
z
x
,
y
X
x
Y
y
2
0
2
0
0
2
2
2
2
2
0
0
z
y
x
z
z
xy
E
h
E
h
E
h
2
2
k
2
y
of
function
x
of
function
0
0
(*)
0
1
1
0
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2 2 2y
Y
k
y
d
y
Y
d
x
X
k
x
d
x
X
d
h
y
d
y
Y
d
y
Y
x
d
x
X
d
x
X
y
Y
x
X
h
y
d
y
Y
d
x
X
y
Y
x
d
x
X
d
y
x
k
k
h
x y2. for (*) to satisfy boundary conditions
E
0
z
(
0
,y)=E
0
z
(a,y)=E
0
z
(x,
0
)=E
0
z
(x,b)=
0
2
2
2
2
2
0
0
(
,
)
sin
sin
,....
3
,
2
,
1
,
,
)
sin(
)
(
,
)
sin(
)
(
b
n
a
m
y
x
mn
b
n
a
m
z
b
n
y
y
a
m
x
x
k
k
h
y
x
E
y
x
E
n
m
k
y
k
y
Y
k
x
k
x
X
Let
m
&
n
define TM
mnwaveguide mode
y
b
n
x
a
m
E
a
m
h
j
x
E
h
j
H
y
b
n
x
a
m
E
b
n
h
j
y
E
h
j
H
y
b
n
x
a
m
E
b
n
h
y
E
h
E
y
b
n
x
a
m
E
a
m
h
x
E
h
E
z
y
z
x
z
y
z
x
sin
cos
cos
sin
cos
sin
sin
cos
0
2
0
2
0
0
2
0
2
0
0
2
0
2
0
0
2
0
2
0
Note
:
h
2
2
k
2
2
h
2
k
2
h
2
2
2
2
2
2
2
2
1
)
(
j
h
jk
f
f
mn
c
b
n
a
m
mn
mn
mn
mn
where
:
cutoff
frequency
of
mn
mode
b
n
a
m
c
TM
f
mn
2
2
2
1
EX:
f
?
0
GHz
)
mn
c
(if
f
1
where
Physical meaning of
cutoff frequency
(
)
mn
c
c
f
f
or
(截止頻率)
(a) if
mn
c
f
f
: (
f
=
input wave
frequency
)
t
z
j
t
j
z
c
mn
mn
f
f
f
f
c
mn
mn
mn
mn
mn
mn
c
mn
c
mn
e
e
e
f
f
jk
j
f
f
jk
j
~
wave
TM
&
imaginary
pure
2
0
1
2
)
(
1
)
(
1
)
(
2
TEM
c
TEM
g
p
TEM
TEM
c
TEM
c
p
u
f
f
u
d
d
u
C
u
C
u
k
u
f
f
k
u
f
f
k
u
2
1
0
0
2
2
1
...
)
1
)
1
/
(
)
(
1
)
/
(
)
(
1
velocity
group
why?)
!
light
of
(speed
medium,
space
-free
(if
*
velocity
phase
=>
u
p
u
g
u
TEM
2
* see 13-3-4
(P
528differential propagation delay or pulse spreading:
min
max
(
1
/
)
)
/
1
(
u
g
u
g
t
)
2
2
1
1
2
2
f
f
f
f
k
c
c
g
:
guide
the
in
wavelength
(
)
1
1
1
2
2
0
0
0
0
TEM
c
c
x
y
y
x
TM
Z
f
f
f
f
k
j
j
j
H
E
H
E
Z
:
impedance
wave
TM
H-field:
a
E
Z
H
z
TM
ˆ
1
(b)
if
mn
c
f
f
(
input wave frequency < cutoff frequency
)
modes
g
propagatin
-non
or
modes
evanescent
:
~
wave
TM
&
real
t
j
z
t
j
z
mn
mn
f
f
f
f
c
mn
mn
mn
e
e
e
e
f
f
jk
j
c
mn
c
mn
mn
1
(
)
2
1
(
)
2
0
* The waveguide acts like a
high-pass filter
f
f
c
f
f
c
Evanescent
13-3-2
TE modes in Rectangular Waveguides
x
z
y
b
a
z
H
z
ˆ
longitudinal components
a H
z
z
a H
z
0
z
x y e
,
z
0
a E
z
z
0
(*) =>
2
xy
H
0
z
h
2
H
0
z
0
h
2
2
k
2
&
y
H
h
x
E
j
y
H
h
H
x
H
h
y
E
j
x
H
h
H
x
H
h
j
x
H
j
y
E
h
E
y
H
h
j
y
H
j
x
E
h
E
z
z
z
y
z
z
z
x
z
z
z
y
z
z
z
x
0
2
0
0
2
0
0
2
0
0
2
0
0
2
0
0
2
0
0
2
0
0
2
0
1
1
1
1
F
rom boundary conditions of the waveguide
tangential
E
-
field
=
0
:
E
0
z
(
0
,y
)
=E
0
z
(
a,y
)
=E
0
z
(
x,
0
)
=E
0
z
(
x,b
)
=
0
since
z
z
z
y
y
z
y
z
z
x
x
z
x
H
x
y
a
H
x
y
H
=
a,y
=E
,y
E
x
H
h
j
E
y
b
x
H
y
x
H
=
x,b
=E
x,
E
y
H
h
j
E
0
0
0
0
0
0
2
0
0
0
0
0
0
2
0
0
)
,
(
)
,
0
(
0
)
(
)
0
(
0
)
,
(
)
0
,
(
0
)
(
)
0
(
for
B.C.
=> determine
H
0
z
x
,
y
&
eigenvalue
h
2
=> then
(
E
0
x
,
E
0
y
,
H
0
x
,
H
0
y
)
1.
By using separation of variables
H
0
z
x
,
y
X
x
Y
y
2
0
2
0
0
2
2
2
2
2
0
0
z
y
x
z
z
xy
H
h
H
h
H
h
2
2
k
2
y
of
function
x
of
function
0
0
(*)
0
1
1
0
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2 2 2y
Y
k
y
d
y
Y
d
x
X
k
x
d
x
X
d
h
y
d
y
Y
d
y
Y
x
d
x
X
d
x
X
y
Y
x
X
h
y
d
y
Y
d
x
X
y
Y
x
d
x
X
d
y
x
k
k
h
x y2. for (*) to satisfy boundary conditions
0(
0
,
)
0(
,
)
0(
,
0
)
0(
,
)
0
y
b
x
H
y
x
H
x
y
a
H
x
y
H
z z z z
2
2
2
2
2
0
0
(
,
)
cos
cos
,....
3
,
2
,
1
,
,
)
cos(
)
(
,
)
cos(
)
(
b
n
a
m
y
x
mn
b
n
a
m
z
b
n
y
y
a
m
x
x
k
k
h
y
x
H
y
x
H
n
m
k
y
k
y
Y
k
x
k
x
X
Let
m
&
n
define TE
mnwaveguide mode
y
b
n
x
a
m
b
n
h
H
y
x
H
y
b
n
x
a
m
a
m
h
H
y
x
H
y
b
n
x
a
m
a
m
h
j
H
y
x
E
y
b
n
x
a
m
b
n
h
j
H
y
x
E
y
x
y
x
sin
cos
,
cos
sin
,
cos
sin
,
sin
cos
,
2
0
0
2
0
0
2
0
0
2
0
0
Note
:
h
2
2
k
2
2
h
2
k
2
h
2
2
h
2
2
2
2
2
jk
1
f
f
2
mn
c
b
n
a
m
mn
mn
where
f
c
mn
(
m
a
)
2
n
b
2
:
cutoff
frequency
of
TE
mn
mode
2
1
Physical meaning of cutoff frequency
(
截止頻率
)
(a) if
mn
c
f
f
: (
f
=
frequency of input wave
,
f
c
=
cutoff frequency
)
t
z
j
mn
mn
f
f
c
mn
mn
mn
mn
mn
c
mn
e
j
f
f
jk
j
~
wave
TE
&
imaginary
pure
0
1
2
2
1
TEM
c
TEM
g
TEM
c
TEM
p
u
f
f
u
d
d
u
u
f
f
u
u
2
1
2
1
1
1
velocity
group
(why?)
velocity
phase
=>
u
p
u
g
u
TEM
2
* see 13-3-4
(P
528differential propagation delay or pulse spreading:
min
max
(
1
/
)
)
/
1
(
u
g
u
g
t
)
2
2
1
1
2
2
f
f
f
f
k
c
c
g
:
guide
the
in
wavelength
)
(
)
(
1
)
(
1
2
1
2
0
0
0
0
TEM
c
c
x
y
y
x
TE
Z
f
f
f
f
k
j
j
j
H
E
H
E
Z
:
impedance
wave
TE
a
E
Z
H
z
TE
ˆ
1
(b)
if
f
f
c
mn
(
input wave frequency < cutoff frequency
)
modes
ting
nonpropaga
or
modes
evanescent
:
~
wave
TE
&
real
t
j
z
mn
mn
f
f
c
mn
mn
mn
e
e
f
f
jk
j
c
mn
mn
1
2
1
2
0
* The waveguide acts like a
high-pass filter
f
f
c
f
f
c
Evanescent
13-3-3
Modal Hierarchy and the Dominant Range
For
TM modes
:
E
0
x
,
y
E
0
sin(
x
)
sin(
y
)
b
n
a
m
z
...
=>
m
n
0
E
0
~
in
(
x
)
sin(
y
)
0
b
n
a
m
z
then
,
or
If
=>
no wave exist
=> So TM
11mode
is the
lowest TM mode
(lowest cutoff frequency)
2
2
1
2
2
2
2
2
1
1
2
2
2
1
)
/
1
(
)
/
1
(
2
)
/
(
)
/
(
2
)
/
1
(
)
/
1
(
)
/
(
)
/
(
11 11 11b
a
b
n
a
m
f
u
b
a
b
n
a
m
f
n
m
TM
c
p
TM
c
n
m
TM
c
For
TE modes
:
H
x
y
H
x
y
b
n
a
m
z
,
0
cos
cos
0
...
=> either
m
or
n
can be
zero
=>
if a > b , TE
10has the
lowest cutoff frequency
a
b
n
a
m
f
u
a
b
n
a
m
f
n
m
c
p
TE
c
n
m
TE
c
TE2
)
/
(
)
/
(
2
1
)
/
(
)
/
(
0
,
1
2
2
2
1
0
,
1
2
2
2
1
10 10 10=>
if a< b , TE
01has the lowest cutoff frequency
b
b
f
TE
c
TE
c
2
1
01
01
2
1
TE
a
b
TE
a
b
10
01
if
if
is the
dominant mode
of a rectangular waveguide .
1
-13
Table
4,
-13
3
-13
EX.
:
range
Usable
*
:
range
Dominant
*
20 10 20 1095
.
0
25
.
1
)
?
(
)
57
.
6
(
c c c cf
f
f
GHz
f
f
GHz
f
dominant
mode
13-3-4 Properties of Propagating Waveguide Modes
Wave Velocities & Dispersion
Transmitted Power
13-3-5 Properties of Modes Below Cutoff
13-3-6 Losses in Metal Waveguides
1
1
2
?
dB/m
)
/
(
1
]
)
/
)(
/
2
(
1
[
2
2
2
10
a
b
b
f
f
b
f
f
a
b
R
c
g
c
g
c
c
s
c
TE
13-3-7 Waveguide Couplers
Fgi. 13-19, 13-20
Coaxial-to-Waveguide Coupler: TE
10mode
13-3-8 (Waveguide) Mode Filters
13-4 Dielectric Waveguides
13-4-1 The Dielectric Slab Waveguide
13-4-2 Fiber-Optic Waveguide (
光纖導波管
)
Multimode & Single-Mode Fibers
o
)
(
2 2 rn
)
(
1 1 rn
)
(
1
0
n
total reflection
cladding
fiber core
多模光纖
單模光纖
Dielectric slab waveguide for
Integrated optical-electronics
