2.3 Rectangular Waveguide—An Artificial Zero-Index System
29
(a) A rectangular waveguide
0
100
200
300
400
500
600
k (m
-1 )
0
2
4
6
8
10
12
14
16
(rad/s)
10
10
10
10
= 3x10
10 rad/s
Waveguide
Air
(b) Dispersion
Fig. 2.2 a Design of a rectangular waveguide and b its dispersion curve indicating zero phase at
the cutoff frequency ω 10 , corresponding to TE 10 mode
where
ω mn = cπ
m
a
2 +
n
b
2
(2.3)
is the cutoff frequency. The wave velocity or the phase velocity is given as
v p =
ω
k
=
c
1 − (ω mn /ω) 2
(2.4)
and the group velocity is given by
v g =
1
dk/dω
= c
1 − (ω mn /ω) 2
(2.5)
Below the cutoff frequency (ω < ω mn ), no mode exists as k is imaginary, while
for very high frequencies (ω >> ω mn ), propagation becomes similar to that in free
space. At very high frequencies, the wavelength λ mn is very small compared to the
dimensions a and b so much so that the incoming wave feels as if it is traveling in free
space. The lowest cutoff frequency is ω 10 , corresponding to TE 10 mode. Figure 2.2b
shows the dispersion curve for the TE 10 mode, along with the dispersion of free space
(or air). It is visible that as the frequency increases, the dispersion approaches freespace behavior. The curve has been plotted for a waveguide of dimension a = 3.0 cm
and b = 2.0 cm, whose lowest cutoff frequency ω 10 = 3 × 10
10 rad/s. Below ω 10 no
mode exists, and all the higher modes have cutoff frequencies higher than ω 10 .
For any mode, at ω = ω mn the wave vector k is zero. Hence, the effective refractive
index is zero, while the phase velocity is infinite. Please note that no laws of physics
29
(a) A rectangular waveguide
0
100
200
300
400
500
600
k (m
-1 )
0
2
4
6
8
10
12
14
16
(rad/s)
10
10
10
10
= 3x10
10 rad/s
Waveguide
Air
(b) Dispersion
Fig. 2.2 a Design of a rectangular waveguide and b its dispersion curve indicating zero phase at
the cutoff frequency ω 10 , corresponding to TE 10 mode
where
ω mn = cπ
m
a
2 +
n
b
2
(2.3)
is the cutoff frequency. The wave velocity or the phase velocity is given as
v p =
ω
k
=
c
1 − (ω mn /ω) 2
(2.4)
and the group velocity is given by
v g =
1
dk/dω
= c
1 − (ω mn /ω) 2
(2.5)
Below the cutoff frequency (ω < ω mn ), no mode exists as k is imaginary, while
for very high frequencies (ω >> ω mn ), propagation becomes similar to that in free
space. At very high frequencies, the wavelength λ mn is very small compared to the
dimensions a and b so much so that the incoming wave feels as if it is traveling in free
space. The lowest cutoff frequency is ω 10 , corresponding to TE 10 mode. Figure 2.2b
shows the dispersion curve for the TE 10 mode, along with the dispersion of free space
(or air). It is visible that as the frequency increases, the dispersion approaches freespace behavior. The curve has been plotted for a waveguide of dimension a = 3.0 cm
and b = 2.0 cm, whose lowest cutoff frequency ω 10 = 3 × 10
10 rad/s. Below ω 10 no
mode exists, and all the higher modes have cutoff frequencies higher than ω 10 .
For any mode, at ω = ω mn the wave vector k is zero. Hence, the effective refractive
index is zero, while the phase velocity is infinite. Please note that no laws of physics