ECE 6341
Spring 2014
Prof. David R. Jackson
ECE Dept.
Notes 6
1
Leaky Modes
v
TM1 Mode
v
R
1
r
u tan u
R  (k0 h) n12  1
SW
u
Splitting point
ISW
f > fs
f = fs
We will examine the solutions as
the frequency is lowered.
2
Leaky Modes (cont.)
v
a) f > fc
SW+ISW
TM1 Mode
SW
The TM1 surface wave is
above cutoff. There is also an
improper TM1 SW mode.
u
ISW
Note: There is also a TM0 mode, but this is not shown
3
Leaky Modes (cont.)
a) f > fc
SW+ISW
The red arrows indicate the direction of
movement as the frequency is lowered.
Im kz
ISW
SW
k0
k1
Re kz
v
TM1 Mode
kz  k0
v  h (k z2  k02 )1/2
  k z2  k02
SW
u
kz  k0
k z  k0
ISW
4
Leaky Modes (cont.)
v
b) f = fc
TM1 Mode
u
The TM1 surface
wave is now at
cutoff. There is
also an improper
SW mode.
5
Leaky Modes (cont.)
b) f = fc
Im kz
k0
k1
Re kz
v
TM1 Mode
u
6
Leaky Modes (cont.)
c) f < fc
2 ISWs
v
TM1 Mode
u
The TM1 surface wave is now
an improper SW, so there are
two improper SW modes.
7
Leaky Modes (cont.)
c) f < fc
2 ISWs
Im kz
k0
k1
Re kz
v
TM1 Mode
The two improper SW
modes approach each
other.
u
8
Leaky Modes (cont.)
d) f = fs
v
TM1 Mode
u
The two improper SW
modes now coalesce.
9
Leaky Modes (cont.)
d) f = fs
Im kz
Splitting point
k0
k1
Re kz
v
TM1 Mode
u
10
Leaky Modes (cont.)
v
e) f < fs
u
The wavenumber kz
becomes complex (and
hence so do u and v).
The graphical solution fails! (It cannot show us complex leaky-wave modal solutions.)
11
Leaky Modes (cont.)
e) f < fs
2 LWs
This solution is rejected as
completely non-physical since it
grows with distance z.
kz   z  j z
Im kz
k0
LW
k1
Re kz
LW
The growing solution is the complex
conjugate of the decaying one (for a
lossless slab).
12
Leaky Modes (cont.)
Proof of conjugate property (lossless slab)
TRE:
1
2 2
z
1
2 2
0
1
 2

2 2 
r 
tan  (k1  k z ) h  

 
(k z2  k )
(k12  k )
TMx Mode
Take conjugate of both sides:
1
*2 2
z
1
2 2
0
 2 *2 12  
r 
tan  (k1  k z ) h  

 
(k z*2  k )
(k12  k )
Hence, the conjugate is a valid solution.
13
Leaky Modes (cont.)
Im kz
SW
ISW
k0
k1
a) f > fc
Re kz
Here we see a
summary of the
frequency behavior for
a typical surface-wave
mode (e.g., TM1).
Im kz
k0
k1
b) f = fc
Re kz
Im kz
k0
c) f < fc
k1
Exceptions:
Re kz
Im kz
splitting point
k0
d) f = fs
k1
Re kz
Im kz
e) f < fs
k0
LW
LW
k1
 TM0: Always remains a
proper physical SW
mode.
 TE1: Goes from proper
physical SW to
nonphysical ISW; remains
nonphysical ISW down to
zero frequency.
Re kz
14
Leaky Modes (cont.)
A leaky mode is a mode that has a complex wavenumber (even for a
lossless structure). It loses energy as it propagates due to radiation.
ˆ x  zk
ˆ z   xˆ x  zˆ z
  Re  k   Re  xk
x

tan0   z /  x
Power flow
0
z
kz   z  j z
15
Leaky Modes (cont.)
One interesting aspect: The fields of the leaky mode must be improper
(exponentially increasing).
Proof:
1
2 2
z
k x 0  (k  k )
2
0
Notes:
z > 0 (propagation in +z direction)
z > 0 (propagation in +z direction)
x > 0 (outward radiation)
kx20  k02  kz2
  x  j x   k02    z  j z 
2
2
x2   x2  j 2x x  k02  z2   z2  j 2 z z
Taking the imaginary part of both sides:
 x x   z  z
16
Leaky Modes (cont.)
For a leaky wave excited by a source, the exponential growth will only
persist out to a “shadow boundary” once a source is considered.
This is justified later in the course by an asymptotic analysis:
In the source problem, the LW pole is only captured when the observation point lies within
the leakage region (region of exponential growth).
x
0
Power flow
z
Source

Region of
weak fields
Leaky mode
Region of
exponential
growth
kz   z  j z
A hypothetical source launches a leaky wave going in one direction.
17
Leaky Modes (cont.)
A leaky-mode is considered to be “physical” if we can measure a
significant contribution from it along the interface (0 = 90o) .
A requirement for a leaky mode to be strongly “physical” is that the
wavenumber must lie within the physical region (z = Re kz < k0) where
is wave is a fast wave*.
(Basic reason: The LW pole is not captured in the complex plane in
the source problem if the LW is a slow wave.)
* This is justified by asymptotic analysis, given later.
18
Leaky Modes (cont.)
f) f < fp
Physical LW
Im kz
k0
Physical
k1
LW
Re kz
Non-physical
f = fp
Note: The physical region is
also the fast-wave region.
Physical leaky wave region (Re kz < k0)
19
Leaky Modes (cont.)
If the leaky mode is within the physical (fast-wave) region, a wedgeshaped radiation region will exist.
This is illustrated on the next two slides.
20
Leaky Modes (cont.)
x

0
Power flow
z
Source
Leaky mode
  xˆ  x  zˆ  z
kz   z  j z
 z   sin  0
2
   x2   z2  k x2  k z2  k02
Hence
z  k0 sin 0
(assuming small attenuation)
Significant radiation requires z < k0.
21
Leaky Modes (cont.)
x

0
Power flow
z
Source
Leaky mode
kz   z  j z
0  sin1  z / k0 
As the mode approaches a slow wave (z  k0), the leakage region
shrinks to zero (0  90o).
22
Leaky Modes (cont.)
Phased-array analogy
x
0
d
z
I0
I1
In
 j  k0d sin0  n
I n  I0 e
IN
n    k0d sin 0  n
Equivalent phase constant:
e jkz z
z  nd
e
 j  k0 d sin 0  n
kz  k0 sin 0
Note:
A beam pointing at an
angle in “visible space”
requires that kz < k0.
23
Leaky Modes (cont.)
The angle 0 also forms the boundary between regions where the leakywave field increases and decreases with radial distance  in cylindrical
coordinates (proof omitted*).
x
Fields are
increasing
radially
Power flow
0
Fields are
decreasing
radially

Source

z
Leaky mode
kz   z  j z
*Please see one of the homework problems.
24
Leaky Modes (cont.)
Excitation problem:
Line source
The aperture field may strongly resemble the field of the leaky
wave (creating a good leaky-wave antenna).
Requirements:
1)
2)
3)
The LW should be in the physical region.
The amplitude of the LW should be strong.
The attenuation constant of the LW should be small.
A non-physical LW usually does not contribute significantly to the aperture field
(this is seen from asymptotic theory, discussed later).
25
Leaky Modes (cont.)
Summary of frequency regions:
a) f > fc
physical SW (non-radiating, proper)
b) fs < f < fc
non-physical ISW (non-radiating, improper)
c) fp < f < fs
non-physical LW (radiating somewhat, improper)
d) f < fp
physical LW (strong focused radiation, improper)
The frequency region fp < f < fc is called the “spectral-gap” region
(a term coined by Prof. A. A. Oliner).
The LW mode is usually considered to be nonphysical in the spectral-gap region.
26
Leaky Modes (cont.)
Spectral-gap region
f = fc
fp < f < fc
Im kz
k0
ISW
f = fs
k1
Re kz
Physical
LW
Non-physical
f = fp
27
Field Radiated by Leaky Wave
x
TEx leaky wave
Aperture
z
Line source
For x > 0:
1
Ey  x, z  
2


Ey  0, kz  e jkx x e jkz z dkz ,

k x   k02  k z2 
1/2
Assume:
E y  0, z   e
 jk zLW z

LW
k
z
Then E y  0, k z   2 j 
 k 2   k LW 2
z
 z




Note: The wavenumber kx is
chosen to be either positive
real or negative imaginary.
28
x x/ /0
k
LW
z
3
/ k0 
 j 0.02
2
Radiation occurs at 60o.
0
10
10
5
5
z / 0
00
-10
-10
0
0
10
10
29
x x/ /0
k
LW
z
3
/ k0 
 j 0.002
2
Radiation occurs at 60o.
0
10
10
5
5
z / 0
00
-10
-10
0
0
10
10
30
x x/ /0
kzLW / k0  1.5  j0.02
The LW is nonphysical.
0
10
10
5
5
z / 0
00
-10
-10
0
0
10
10
31
Leaky-Wave Antenna
x
x / 0
Aperture
z
Line source
x x/ /
0
0
10
10
Near field
Far field
5
5
z / 0
00
-10
0
10
32
Leaky-Wave Antenna (cont.)
x
x / 0


z
line source
Far-Field Array Factor (AF)
AF   


E y  0, z  e
 j  k0 sin   z
dz



e
 jk zLW z
e
 j  k0 sin   z
dz



LW
k
z

AF    2 j 
2
 k 2 sin 2    k LW  
z
 0

33
Leaky-Wave Antenna (cont.)


 z  j z

AF    2 j  2 2
2
 k sin      j  
z
z
 0


 z  j z

AF    2 2 2
 k0 sin    z2   z2  j 2 z  z





1/2


2
2



z
z

AF    2 
2
  k 2 sin 2    2   2    2  2 
z
z
z z
 0

A sharp beam occurs at
k0 sin 0   z
34
Leaky-Wave Antenna (cont.)
Two-layer
x /  (substrate/superstrate) structure excited by a line source.
0
x
r2

t
 r1
z
b
Far Field
 r 2   r1
b / 1  0.5
t / 2  0.25
Note: Here the two beams have merged to become a single beam at broadside.
35
Leaky-Wave Antenna (cont.)
x / 0
Implementation at millimeter-wave frequencies (62.2 GHz)
r1 = 1.0, r2 = 55, h = 2.41 mm, t = 0.484 mm, a = 3.73 0 (radius)
36
Leaky-Wave Antenna (cont.)
x / 0
(E-plane shown on one side, H-plane on the other side)
37
Leaky-Wave Antenna (cont.)
•W. W. Hansen, “Radiating electromagnetic waveguide,” Patent, 1940, U.S. Patent No. 2.402.622.
W. W. Hansen, “Radiating electromagnetic waveguide,” Patent, 1940, U.S. Patent No. 2.402.622.
y
This is the first leaky-wave antenna invented.
y
Slotted waveguide
x
 
 z  k02   
a
z
b

a
Note:
2

 z  k0
38
Leaky-Wave Antenna (cont.)
x / 0
The slotted waveguide illustrates in a simple way why the field
is weak outside of the “leakage region.”
Region of strong fields (leakage region)
Slot
a
TE10 mode
Waveguide
Source
Top view
39
Leaky-Wave Antenna (cont.)
x / 0
Another variation: Holey waveguide
y
y
x
z
 
r
b
a
2
 z  k02   
a

p

p  0
40
Leaky-Wave Antenna (cont.)
x / 0
Another type of leaky-wave antenna: Surface-Integrated Waveguide (SIW)
s
ls
w
ws
via
d
p
slot
h
εr
41