### Notes 12 - Surface waves

```ECE 5317-6351
Microwave Engineering
Fall
Fall2011
2011
Prof. David R. Jackson
Dept. of ECE
Notes 12
Surface Waves
1
Grounded Dielectric Slab
Discontinuities on planar transmission lines such as microstrip
Microstrip line
Substrate
(ground plane below)
Surface-wave field
It is important to understand these waves.
Note: Sometimes layers are also used as a desired propagation mechanism for microwave
and millimeter-wave frequencies. (The physics is similar to that of a fiber-optic guide.)
2
Grounded Dielectric Slab
x
h
 r , r
z
Goal: Determine the modes of propagation and their wavenumbers.
Assumption: There is no variation of the fields in the y direction,
and propagation is along the z direction.
3
Dielectric Slab
TMx & TEx modes:
Note: These modes may also be
classified as TMz and TEz.
x
TMx
H
Plane wave
E
 r , r
z
x
TEx
E
H
Plane wave
 r , r
z
4
Surface Wave
Exponential decay
x
1
Plane wave
 r , r
z
1  c
The internal angle is greater than the critical angle, so there is
exponential decay in the air region.
kz  k1 sin1  k0
The surface wave is a “slow wave”.
Hence
kx0   k  k
2
0

2 1/2
z
  j k z2  k02   j x 0
5
TMx Solution
Assume TMx
Ex  e jkz z ex  x 
 2 Ex Ex  2 Ex
2



k
Ex  0
2
2
2
x
y
z
 2 Ex
2
2

k

k

z  Ex  0
2
x
6
TMx Solution (cont.)
 2ex  x 
2
2

k

k
ex  x   0


z
2
x
k x 0   k02  k z2 
1/2
Denote
  j k z2  k02   j x 0
k x1  k12  k z2
Then we have
xh
 2ex  x 
2

k
x 0 ex  x   0
2
x
xh
 2ex  x 
2

k
x1ex  x   0
2
x
7
TMx Solution (cont.)
Applying boundary conditions at the ground plane, we have:
xh
Ex1  e
xh
Ex 0  Ae jkz z e x 0 x
 jkz z
Note:
cos(kx1x)
En
Ex
0 
0
n
x
This follows since
  D  v  0
x
h
Et  0
 r , r
z
8
Boundary Conditions
BC 1)
Dx0  Dx1
@x  h
 Ex 0   r Ex1
BC 2)
Ez 0  Ez1 @ x  h

E x 0 E x1

x
x
Note:
Ex Ey Ez
Ex
 E  0 


0 
    jk z  Ez
x
y
z
x
9
Boundary Conditions (cont.)
These two BC equations yield:
Ae x 0h   r cos(k x1h)
 x 0 Ae x 0h  (k x1 ) sin(k x1h)
Divide second by first:
 x 0 
1
r
(k x1 ) tan(k x1h)
or
 x0  r  kx1 tan(kx1h)
10
Final Result: TMx
This may be written as:
 r kz 2  k02  k12  kz 2 tan  k12  kz 2 h


This is a transcendental equation for the unknown wavenumber kz.
11
Final Result: TEx
Omitting the derivation, the final result for TEx modes is:
k z 2  k0 2  
1
r
k12  k z 2 cot

k12  k z 2 h

This is a transcendental equation for the unknown wavenumber kz.
12
Graphical Solution for SW Modes
 x0 r  kx1 tan(kx1h)
Consider TMx:
 x0h 
or
Let
1
r
(k x1h) tan(k x1h)
u  k x1h
v   x0h
Then
v
1
r
u tan u
13
Graphical Solution (cont.)
We can develop another equation by relating u and v:
u  h k12  k z 2
v  h k z 2  k0 2
Hence
u 2  h 2 (k12  k z 2 )
v  h ( k z  k0 )
2
2
u  v  h (k  k0 )
2
2
2
2
1
2
 (k0 h) (n  1)
2
2
1
2
2
where
n1  k1 / k0   r r
14
Graphical Solution (cont.)
Define
R2  (k0h)2 (n12 1)
R  ( k0 h) n  1
2
1
Note: R is proportional
to frequency.
Then
u v  R
2
2
2
15
Graphical Solution (cont.)
Summary for TMx Case
v
1
r
u tan u
u v  R
2
2
2
u  k x1h  h k12  k z2
v   x 0 h  h k z2  k02
16
Graphical Solution (cont.)
v
v
TM0
1
r
u tan u
R
p /2
p
3p /2
u
u 2  v2  R2
R  (k0 h) n12  1
17
Graphical Solution (cont.)
v
Graph for a Higher Frequency
TM0
TM1
R
p /2
p
3 p /2
u
Improper SW (v < 0)
(We reject this solution.)
18
Proper vs. Improper
v   x0h
If v > 0 : “proper SW” (fields decrease in x direction)
If v < 0 : “improper SW” (fields increase in x direction)
Cutoff frequency: The transition between a proper and improper mode.
Note: This definition is different from that for a closed waveguide
structure (where kz = 0 at the cutoff frequency.)
Cutoff frequency: TM1 mode:
v0
u p
19
TMx Cutoff Frequency
v
R p
TM1:
k0 h n12  1  p

p
u
h
0
1/ 2

n12  1
R
For other TMn modes:
TM n :
h
0

n/2
n12  1
n  0,1, 2,...
20
TM0 Mode
The TM0 mode has no cutoff frequency (it can propagate at
any frequency:
Note: The lower the frequency, the more loosely bound the field is in
the air region (i.e., the slower it decays away from the interface).
kz / k 0
 x 0  k z2  k02
n1
TM0
TM1
1.0
fcTM1
f
21
TM0 Mode
After making some approximations to the transcendental equation,
valid for low frequency, we have the following approximate result for
the TM0 mode:
  k h   n2  1
0
1

 k0 1 

 r2

2
TM
0
2
1/ 2




k0 h  1
22
TEx Modes
v
TE1
TE2
 x0h  
1
r
 kx1h  cot  kx1h 
R
p/2
1/ r
Hence
p
3p/2
u
v
1
r
u cot u
23
TEx Modes (cont.)
No TE0 mode ( fc= 0). The lowest TEx mode is the TE1 mode.
TE1 cut-off frequency at (R  p / 2 :
 k0 h 
h
0
n 1 
2
1

p
2
1/ 4
n12  1
In general, we have
TEn:
h
0
2n  1 / 4


n12  1
n  1, 2,3,.........
The TE1 mode will
start to propagate
when the substrate
thickness is roughly
1/4 of a dielectric
wavelength.
24
TEx Modes (cont.)
Here we examine the radiation efficiency er of a small electric dipole placed
on top of the substrate (which could mode a microstrip antenna).
TM0 SW
100
EFFICIENCY (%)
80
 r  2.2
60
 r  10.8
40
exact
0
er 
20
0
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
hh//  00
Psp
Psp  Psw
0.1
25
Dielectric Rod
z
a
 r , r
The physics is
similar to that of
the TM0 surface
wave on a layer.
This serves as a model for a single-mode fiber-optic cable.
26
Fiber-Optic Guide (cont.)
Two types of fiber-optic guides:
1) Single-mode fiber
This fiber carries a single mode (HE11). This requires the fiber diameter to
be on the order of a wavelength. It has less loss, dispersion, and signal
distortion. It is often used for long-distances (e.g., greater than 1 km).
2) Multi-mode fiber
This fiber has a diameter that is large relative to a wavelength (e.g., 10
wavelengths). It operates on the principle of total internal reflection
(critical-angle effect). It can handle more power than the single-mode
fiber, but has more dispersion.
27
Dielectric Rod (cont.)
Dominant mode (lowest cutoff frequency): HE11
(fc = 0)
The dominant mode is a hybrid
mode (it has both Ez and Hz).
E
Single-mode fiber
Note: The notation HE means that the mode is hybrid, and has both Ez and Hz,
although Hz is stronger. (For an EH mode, Ez would be stronger.)
The field shape is somewhat similar to the TE11 waveguide mode.
The physical properties of the fields are similar to those of the TM0 surface wave on
a slab (For example, at low frequency the field is loosely bound to the rod.)
28
Dielectric Rod (cont.)
http://en.wikipedia.org/wiki/Optical_fiber
What they look like in practice:
Single-mode fiber
Fiber-Optic Guide (cont.)
Higher index core region
Multimode fiber
A multimode fiber can be
explained using
geometrical optics and
internal reflection. The
“ray” of light is actually a
superposition of many
waveguide modes (hence
the name “multimode”).
http://en.wikipedia.org/wiki/Optical_fiber
30
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