6341 notes 36 Poles and Branch Cuts

Report
ECE 6341
Spring 2014
Prof. David R. Jackson
ECE Dept.
Notes 36
1
Radiation Physics in Layered Media
y
Line source on grounded substrate
I0
x
r
h
Note: TMz and also TEy (since
For y > 0:
0 I 0

4 j

 0)
z
E  zˆ Ez  x, y 
  Az



1
1  TE  k x   e  jk y 0 y e  jkx x dk x
k y0
2
Reflection Coefficient
TE  k x  
ZinTE  k x   Z 0TE  k x 
ZinTE  k x   Z 0TE  k x 
TE
Z00
I
where
Z
ZinTE  kx   jZ1TE tan  k y1h
TE
0
TE
1
Z


0
k y0
0
k y1
+ V TE
Z01
k y0   k  k
2
0

2 1/ 2
x
k y1   k  k
2
1
z

2 1/ 2
x
3
Poles
TE  k x  
Poles:
ZinTE  k x   Z 0TE  k x 
ZinTE  k x   Z 0TE  k x 
k x  k xp
ZinTE  kxp   Z0TE  kxp 
This is the same equation as the TRE for finding the wavenumber of a surface wave:
ZinTE  kxSW   Z0TE  kxSW 
TE
Z00
I
kxp = roots of TRE = kxSW
+ V -
Z
TE
01
z
4
Poles (cont.)
Complex kx plane
kxi
k x  k xr  jk xi
SW
 k1
C
 k0
k xr
k0
SW
k1
If a slight loss is added, the SW poles are shifted
off the real axis as shown.
5
Poles (cont.)
kxi
 k1
 k0
C
k0
k1
k xr
kxi
 k1
 k0
C
k0
k1
k xr
For the lossless case, two possible paths are shown here.
6
Review of Branch Cuts
and Branch Points
In the next few slides we review the basic concepts of
branch points and branch cuts.
7
Branch Cuts and Points (cont.)
f  z   z1/ 2
Consider
1/ 2
z
Choose
z 1
 r e

j 1/ 2
z  r e j
 r e j / 2
  0:
z1/ 2  1
  2 :
z1/ 2  1
  4 :
z1/ 2  1
There are two possible values.
8
Branch Cuts and Points (cont.)
The concept is illustrated for
1/ 2
z
f  z   z1/ 2
 re
z  r e j
j / 2
y
Consider what happens if we
encircle the origin:
r=1
C
B
A
x
9
Branch Cuts and Points (cont.)
y
z1/ 2  r e j / 2
r=1
C
B
A
point

A
B
C
0

2
x
z1/2
1
+j
-1
We don’t get back the same result!
10
Branch Cuts and Points (cont.)
z1/ 2  r e j / 2
y
Now consider encircling
the origin twice:
r=1
r
D
C
B
A
point

A
B
C
D
E
0

2
3
4
E
x
z1/2
1
+j
-1
-j
1
We now get back the same result!
Hence the square-root function is a
double-valued function.
11
Branch Cuts and Points (cont.)
The origin is called a branch point: we are not allowed to encircle it if
we wish to make the square-root function single-valued.
In order to make the square-root function single-valued, we
must put a “barrier” or “branch cut”.
y
Branch cut
x
Here the branch cut was chosen to lie on the negative real axis (an arbitrary choice)
12
Branch Cuts and Points (cont.)
We must now choose what “branch” of the function we want.
z  r e j
This is the "principle"
branch, denoted by
1/ 2
z
 re
j / 2
y
z
Branch cut
    
z 1
MATLAB :      
x
z1/ 2  1
13
Branch Cuts and Points (cont.)
Here is the other choice of branch.
z  r e j
1/ 2
z
 re
j / 2
y
Branch cut
    3
z 1
x
z1/ 2  1
14
Branch Cuts and Points (cont.)
Note that the function is discontinuous across the branch cut.
z  r e j
1/ 2
z
 re
j / 2
y
Branch cut
z  1,    
    
z1/ 2  j
z  1,    
z1/ 2   j
z 1
x
z1/ 2  1
15
Branch Cuts and Points (cont.)
The shape of the branch cut is arbitrary.
z  r e j
z1/ 2  r e j / 2
y
 / 2    3 / 2
z 1
Branch cut
x
z1/ 2  1
16
Branch Cuts and Points (cont.)
The branch cut does not even have to be a straight line
z  re
j
z1/ 2  r e j / 2
In this case the branch is determined by requiring that the
square-root function (and hence the angle  ) change
continuously as we start from a specified value (e.g., z = 1).
y
z  1
z
1/ 2
 j
z j
z1/ 2  e j / 4  1  j  / 2
z 1
Branch cut
x
z1/ 2  1
z j
z1/ 2  e j / 4  1  j  / 2
17
Branch Cuts and Points (cont.)
Consider this function:
f ( z )   z  1
2
1/ 2
(similar to our wavenumber function)
What do the branch points and branch cuts look like for this function?
18
Branch Cuts and Points (cont.)
f ( z )   z  1
2
1/ 2
  z  1
1/ 2
 z  1
1/ 2
  z  1
1/ 2
 z   1
1/ 2
y
1
x
1
There are two branch cuts: we are not allowed to encircle either branch point.
19
Branch Cuts and Points (cont.)
f ( z)   z  1
Geometric interpretation
1/ 2
 z   1
1/ 2
y
 w11/ 2 w1/2 2
w1  z  1  r1 e j1
w2  z  (1)  r2 e j2
w2
1
2
w1
1
x
1
The function f (z) is unique once we specify
its value at any point. (The function must
change continuously away from this point.)
f ( z) 

r1 e j1 / 2

r2 e j2 / 2

20
Riemann Surface
The Riemann surface is really multiple
complex planes connected together.
The function z1/2 has a surface with
two sheets.
The function z1/2 is continuous
everywhere on this surface (there are
no branch cuts). It also assumes all
possible values on the surface.
Georg Friedrich Bernhard
Riemann (September 17, 1826 –
July 20, 1866) was an influential
German mathematician who made
lasting contributions to analysis
and differential geometry, some of
them enabling the later
development of general relativity.
21
Riemann Surface
The concept of the Riemann surface is illustrated for
f  z   z1/ 2
z  r e j
Consider this choice:
Top sheet:
Bottom sheet:
    
( 1  1)
    3 ( 1  1)
22
Riemann Surface (cont.)
x
Top
y
D
B
Bottom
B
D
y
D
B
x
D
B
side view
top view
23
Riemann Surface (cont.)
Top sheet
Branch point
Branch cut
(where it used to be)
Bottom sheet
24
Riemann Surface (cont.)
y
r=1
Connection between
sheets
D
D
C
B
B
A
point

A
B
C
D
E
0

2
3
4
E
x
z1/ 2
1
+j
-1
-j
1
25
Branch Cuts in Radiation Problem
Now we return to the original problem:
  Az
0 I 0

4 j



1
 jk y 0 y  jk x x
TE
1    k x   e
e
dk x
k y0
k y 0   k0  kx
2
1
2 2

Note: There are no branch points from ky1:
k y1   k  k
2
1
1
2 2
x

ZinTE  kx   jZ1TE tan  k y1h
Z1TE 
0
k y1
26
Branch Cuts
k y 0   k0  k
2
1
2 2
x
  k
  j  k x  k0 
1
2
0
 kx 
 k x  k0 
1
2
 k0  k x 
1
2
1
2
Note: It is arbitrary that we have factored out a –j instead of a +j, since
we have not yet determined the meaning of the square roots.
Branch points appear at
k x   k0
No branch cuts appear at
k x   k1
(The integrand is an even
function of ky1.)
27
Branch Cuts (cont.)
k y 0   j  k x  k0 
1
2
 k x  k0 
1
2
kxi
C
 k1
 k0
k0
k xr
k1
Branch cuts are lines we are not allowed to cross.
28
Branch Cuts (cont.)
For
k x  real  k0 ,
Choose
k y0   j k y0
 arg


 arg

 k x  k0   0
 k x  k0   0
at this point
k y0   k  k
2
0

2 1/ 2
x
k y 0   j  k x  k0 
1
2
kxi
 k x  k0 
1
2
k0
 k0
k xr
k y0   j k y0
This choice then uniquely defines ky0
everywhere in the complex plane.
29
Branch Cuts (cont.)
For
k x  real,
we have arg  k x  k0   
0  k x  k0
arg  k x  k0   0
k y 0   j  k x  k0 e j / 2   k x  k0 



Hence
kxi
ky0  ky0
 k0
k0
k xr
30
Riemann Surface
k y0   k  k
2
0

2 1/ 2
x
Top sheet
kxi
ky0   j ky0
k0
k xr
 k0
ky0   j ky0
Bottom sheet
There are two sheets, joined at the blue lines.
31
Proper / Improper Regions
Let
k x  k xr  jk xi
k0  k0  jk0
k y 0   k02  k
The goal is to figure out which
regions of the complex plane
are "proper" and "improper."
1
2 2
x

“Proper” region:
Im k y 0  0
“Improper” region:
Im k y 0  0
Boundary:
Im k y 0  0
k y20  k02  k x2  real >0
32
Proper / Improper Regions (cont.)
Hence


k0  jk0

2
  k xr  jk xi   real  0
2
 

k02  k02  k xr2  k xi2  j 2k0k0  2k xr k xi  real  0
Therefore
k xr kxi  k0k0
One point on curve:
k xr  k0
k xi  k0
kx  k0  k0  jk0
(hyperbolas)
kxi
k0
k xr
k0
k0  k0  jk0
33
Proper / Improper Regions (cont.)
Also
k02  k02  kxr2  kxi2  0
kxi
The solid curves
satisfy this condition.
k0
k xr
k0
34
Proper / Improper Regions (cont.)
kxi
Complex plane: top sheet
k0
k xr
k0
Proper
Improper region
On the complex plane corresponding to the bottom sheet, the proper and
improper regions are reversed from what is shown here.
35
Sommerfeld Branch Cuts
kxi
k0
k0
k xr
Hyperbola
Complex plane corresponding to top sheet: proper everywhere
Complex plane corresponding to bottom sheet: improper everywhere
36
Sommerfeld Branch Cuts
kxi
kxi
Complex plane
k0
Riemann surface
k0
k xr
k0
k xr
k0
Note: We can think of a single complex plane with branch cuts, or a Riemann surface
with hyperbolic-shaped “ramps” connecting the two sheets.
The Riemann surface allows us to show all possible poles, both proper
(surface-wave) and improper (leaky-wave).
37
Sommerfeld Branch Cut
kxi
Let
k0  0
k0
k xr
k0
The branch cuts now lie along the imaginary axis, and part of the real axis.
38
Path of Integration
kxi
k1
k0
C
k xr
k0
k1
The path is on the complex plane corresponding to the top Riemann sheet.
39
Numerical Path of Integration
kxi
C
k1
k0
k0
k1
k xr
40
Leaky-Mode Poles
TRE:
Review of frequency behavior
Zin  k xLW    Z 0  k xLW 
kxi
Im k y 0  0
(improper)
f 0
f  fc
k0
SW
k1
ISW
Note: TM0 never
becomes improper
LW
k xr
f  fs
Bottom sheet
41
Riemann Surface
We can now show the
leaky-wave poles!
k1
kxi
C
k0
BP
k0
k
LW
xp

LW
 j
SWP
k1
k xr
LWP
LW
 LW  Re  kxpLW 
k0  Re  k
LW
xp
k
0
The LW pole is then “close” to the path
on the Riemann surface (and it usually
makes an important contribution).
42
SW and CS Fields
kxi
SW field
CS field
k0
k1
k xr
Cp
Cb
LW
SW
Total field = surface-wave (SW) field
+ continuous-spectrum (CS) field
Note: The CS field indirectly accounts for the LW pole.
43
Leaky Waves
LW poles may be important if
k0  Re  kxpLW   k0
Im kxp
Physical Interpretation
 LW  k0 sin0
k0
The LW pole is then
“close” to the path on the
Riemann surface.
radiation
0
 LW  Re(kxpLW )
leaky wave
44
Improper Nature of LWs
“leakage rays”
Region of strong
leakage fields
kxpLW    j
The rays are stronger near the beginning of the wave: this gives us
exponential growth vertically.
45
Improper Nature (cont.)
Mathematical explanation of exponential growth (improper behavior):
k
LW
y0
k 
   j 
LW
y0
2
2
y
y
  k0  k
LW 2
xp
 k0   k
LW
xp
2
2


1
2
2
 k0 2     j 
2
Equate imaginary parts:
 y y    
 
 y   

 y
  0  y  0



(improper)
46

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