Prof. David R. Jackson ECE Dept. Spring 2014 Notes 6 ECE 6341 1
Dec 22, 2015
1
Prof. David R. JacksonECE Dept.
Spring 2014
Notes 6
ECE 6341
2
Leaky Modes
f = fs
v
u
TM1 Mode
ISW
SW
f > fs We will examine the solutions asthe frequency is lowered.
Splitting point
1tan
r
v u u
R
20 1( ) 1R k h n
3
Leaky Modes (cont.)
a) f > fc SW+ISW
v
u
ISW
SW
TM1 Mode
The TM1 surface wave is above cutoff. There is also an improper TM1 SW mode.
Note: There is also a TM0 mode, but this is not shown
4
Leaky Modes (cont.)
a) f > fc SW+ISW
Im kz
k1
SW
Re kz
k0
ISW
The red arrows indicate the direction of movement as the frequency is lowered.
2 2 1/20
2 20
( )z
z
v h k k
k k
0zk k
0zk k 0zk k
v
u
ISW
SW
TM1 Mode
5
b) f = fc
Leaky Modes (cont.)
v
u
TM1 Mode
The TM1 surface wave is now at cutoff. There is also an improper SW mode.
6
b) f = fc
Leaky Modes (cont.)
Im kz
k1
Re kz
k0
v
u
TM1 Mode
7
c) f < fc 2 ISWs
Leaky Modes (cont.)
v
u
The TM1 surface wave is now an improper SW, so there are two improper SW modes.
TM1 Mode
8
c) f < fc 2 ISWs
Im kz k1
Re kz
k0
Leaky Modes (cont.)
The two improper SW modes approach each other.
v
u
TM1 Mode
9
d) f = fs
Leaky Modes (cont.)
v
u
The two improper SW modes now coalesce.
TM1 Mode
10
d) f = fs
Leaky Modes (cont.)
Im kz k1
Re kz
k0
Splitting point
v
u
TM1 Mode
11
e) f < fs
Leaky Modes (cont.)v
u
The graphical solution fails! (It cannot show us complex leaky-wave modal solutions.)
The wavenumber kz becomes complex (and hence so do u and v).
12
Leaky Modes (cont.)
Im kz
k1
e) f < fs 2 LWs
Re kz
k0
LW
LW
This solution is rejected as completely non-physical since it grows with distance z.
The growing solution is the complex conjugate of the decaying one (for a lossless slab).
z z zk j
13
Leaky Modes (cont.)
Proof of conjugate property (lossless slab)
Take conjugate of both sides:
112 2 2
2 21 211
2 2 20
( )tan ( )
( )
zr z
z
k kk k h
k k
112 *2 2
2 *21 211
*2 2 20
( )tan ( )
( )
zr z
z
k kk k h
k k
Hence, the conjugate is a valid solution.
TRE: TMx Mode
14
Leaky Modes (cont.)
a) f > fc
Im kz
k1
SW
Re kz
k0
ISW
b) f = fc
Im kz
k1
Re kz
k0c) f < fc
Im kz
k1
Re kz
k0
splitting point
d) f = fs
Im kz
k1
Re kz
k0
LW
LW
e) f < fs
Here we see a summary of the
frequency behavior for a typical surface-wave
mode (e.g., TM1).
TM0: Always remains a proper physical SW mode.
TE1: Goes from proper physical SW to nonphysical ISW; remains nonphysical ISW down to zero frequency.
Exceptions:
Im kz
k1
Re kz
k0
15
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.
z
Power flow
x
z z zk j
ˆ ˆˆ ˆRe Re x z x zk xk zk x z
0
0tan /z x
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Leaky Modes (cont.)
One interesting aspect: The fields of the leaky mode must be improper (exponentially increasing).
Proof:1
2 2 20 0( )x zk k k
2 2 20 0x zk k k
2 220x x z zj k j
2 2 2 2 202 2x x x x z z z zj k j
x x z z Taking the imaginary part of both sides:
Notes:z > 0 (propagation in +z direction)z > 0 (propagation in +z direction)
x > 0 (outward radiation)
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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).
0
z
Power flow
Leaky mode
x
Region of exponential
growth
Region of weak fields
z z zk j Source
A hypothetical source launches a leaky wave going in one direction.
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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*.
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) .
(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.
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f) f < fp Physical LW
Physical leaky wave region (Re kz < k0)
Leaky Modes (cont.)
Im kz
k1
Re kz
k0 LW
f = fp
PhysicalNon-physical
Note: The physical region is also the fast-wave region.
20
If the leaky mode is within the physical (fast-wave) region, a wedge-shaped radiation region will exist.
Leaky Modes (cont.)
This is illustrated on the next two slides.
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0sinz
2 2 2 2 2 20x z x zk k k
0 0sinz k Hence
Leaky Modes (cont.)
(assuming small attenuation)
ˆ ˆx zx z
Significant radiation requires z < k0.
0
z
Power flow
Leaky mode
x
z z zk j Source
22
10 0sin /z k
Leaky Modes (cont.)
0
z
Power flow
Leaky mode
x
z z zk j Source
As the mode approaches a slow wave (z k0), the leakage region shrinks to zero (0 90o).
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Leaky Modes (cont.)
0 0sin0
j k d nnI I e
Phased-array analogy
z
I0
x
I1 In IN
d
0
0 0sinzk k
Note:A beam pointing at an angle in “visible space”
requires that kz < k0.
0 0sinz j k d njk z
z nde e
0 0sinn k d n
Equivalent phase constant:
24
Leaky Modes (cont.)
The angle 0 also forms the boundary between regions where the leaky-wave field increases and decreases with radial distance in cylindrical coordinates (proof omitted*).
0
z
Power flow
Leaky mode
x
Fields are decreasing
radially
z z zk j Source
Fields are increasing
radially
*Please see one of the homework problems.
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The aperture field may strongly resemble the field of the leaky wave (creating a good leaky-wave antenna).
Leaky Modes (cont.)
Line source
Requirements:
1) The LW should be in the physical region.2) The amplitude of the LW should be strong.3) 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).
Excitation problem:
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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.
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Leaky Modes (cont.)
Im kz
k1
Re kz
k0
LW
f = fp
Physical Non-physical
ISW
f = fc
f = fs
Spectral-gap region
fp < f < fc
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Field Radiated by Leaky Wave
1, 0, ,
2x zjk x jk z
y y z zE x z E k e e dk
0,LWzjk z
yE z e
22
0, 2LWz
y zLW
z z
kE k j
k k
For x > 0:
Then
Line sourcez
x
Aperture
1/22 20x zk k k
Note: The wavenumber kx is chosen to be either positive real or negative imaginary.
Assume:
TEx leaky wave
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z / 0
x / 0
-10 100
0
10
0
3/ 0.02
2LWzk k j
5
x / 0
0
10
5
-10 0 10
Radiation occurs at 60o.
30
z / 0
x / 0
-10 100
0
10
0
3/ 0.002
2LWzk k j
5
x / 0
0
10
5
-10 0 10
Radiation occurs at 60o.
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z / 0
x / 0
-10 100
0
10
0/ 1.5 0.02LWzk k j
5
x / 0
0
10
5
-10 0 10
The LW is nonphysical.
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x / 0
Leaky-Wave Antenna
z / 0
x / 0
-10 100
0
10
5
x / 0
0
10
5
Near field
Far field
Line sourcez
x
Aperture
33
Leaky-Wave Antenna (cont.)
Far-Field Array Factor (AF)
0
0
sin
sin
0,
LWz
j k zy
j k zjk z
AF E z e dz
e e dz
22 2
0
2sin
LWz
LWz
kAF j
k k
x / 0
line sourcez
x
34
Leaky-Wave Antenna (cont.)
22 2
0
2sin
z z
z z
jAF j
k j
2 2 2 20
2sin 2
z z
z z z z
jAF
k j
1/22 2
2 22 2 2 20
2sin 2
z z
z z z z
AFk
A sharp beam occurs at 0 0sin zk
35
x / 0
Leaky-Wave Antenna (cont.)Two-layer (substrate/superstrate) structure excited by a line source.
2 1r r
1
2
/ 0.5
/ 0.25
b
t
1r2r
b
t
x
z
Far Field
Note: Here the two beams have merged to become a single beam at broadside.
36
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)
Leaky-Wave Antenna (cont.)
37
x / 0
Leaky-Wave Antenna (cont.)
(E-plane shown on one side, H-plane on the other side)
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Leaky-Wave Antenna (cont.)W. W. Hansen, “Radiating electromagnetic waveguide,” Patent, 1940, U.S. Patent No. 2.402.622.
a
b
x
z
y
220z k
a
0z k Note:
•W. W. Hansen, “Radiating electromagnetic waveguide,” Patent, 1940, U.S. Patent No. 2.402.622.
y This is the first leaky-wave antenna invented.
Slotted waveguide
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x / 0
Leaky-Wave Antenna (cont.)
The slotted waveguide illustrates in a simple way why the field is weak outside of the “leakage region.”
Top view
Source
Slot
Region of strong fields (leakage region)
Waveguidea TE10 mode
40
x / 0
Leaky-Wave Antenna (cont.)Another variation: Holey waveguide
220z k
a
a p
b
x
z
y
r
0p
y
41
x / 0
Leaky-Wave Antenna (cont.)
Another type of leaky-wave antenna: Surface-Integrated Waveguide (SIW)
h εr
via d s
p ls ws w
slot