4/30/2015
ECE 5390 Special Topics:
21st Century Electromagnetics
Instructor:
Office:
Phone:
E‐Mail:
Dr. Raymond C. Rumpf
A‐337
(915) 747‐6958
[email protected]
Spring 2014
Lecture #16
Transformation
Electromagnetics
Lecture 16
1
Lecture Outline
•
•
•
•
•
•
•
•
•
Introduction
Coordinate transformations
Form invariance of Maxwell’s equations
Transformation electromagnetics
Stretching space
Cloaking
Carpet cloaking
Other applications
Concluding remarks
Lecture 16
Slide 2
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Introduction
Design Process Using Spatial
Transforms
Step 1 of 4:
Define Spatial Transform
Lecture 16
4
2
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Design Process Using Spatial
Transforms
Step 2 of 4:
Calculate Effective Material Properties



  E   j H


  H  j E

  E   j H


  H  j E
Lecture 16
5
Design Process Using Spatial
Transforms
Step 3 of 4:
Map Properties to Engineered Materials

Lecture 16
&

6
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Design Process Using Spatial
Transforms
Step 4 of 4:
Generate Overall Lattice
Lecture 16
7
Coordinate
Transformation
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Coordinate Transformation
We can map one coordinate space into another
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
r   xxˆ   yyˆ   z zˆ

r  xxˆ  yyˆ  zzˆ
Lecture 16
9
Jacobian Matrix [A]
To aid in the coordinate transform, we use the Jacobian
transformation matrix.

 A    r  
Gradient of a vector is a tensor! Wow! 
T
 x
 x

 y

 x
 z 

 x
x
y
y
y
z 
y
x 
z 
y 

z 
z  

z 
Each term quantifies the “stretching” of the coordinates.
The Jacobian matrix does not perform a coordinate transformation. It transforms functions and operations between different coordinate systems.
Lecture 16
10
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Example #1: Cylindrical to
Cartesian
The Cartesian and cylindrical coordinates are related through
x   cos 
y   sin 
zz
The Jacobian matrix is then
 x 
 A  y 
 z 
cos 
 A   sin 
 0
x 
y 
z 
  sin 
 cos 
0
x z 
y z 
z z 
0
0 
1 
x
 cos 

y
 sin 

z
0

x
x
   sin 
0

z
y
y
  cos 
0

z
z
z
0
1

z
Lecture 16
11
Example #2: Spherical to
Cartesian
The Cartesian and spherical coordinates are related through
x  r sin  cos 
y  r sin  sin 
z  r cos 
The Jacobian matrix is then
 x r x 
 A  y r y 
 z r z 
sin  cos 
 A   sin  sin 
 cos 
Lecture 16
x  
y  
z  
r cos  cos 
r cos  sin 
r sin 
x
 sin  cos 
r
y
 sin  sin 
r
z
 cos 
r
x
 r cos  cos 

y
 r cos  sin 

z
  r sin 

x
  r sin  sin 

y
 r sin  cos 

z
0

r sin  sin  
r sin  cos  

0
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Example #3: Cartesian to
Cartesian
Coordinates in Cartesian space can be transformed according to
x  x  x, y , z 
y   y   x, y , z 
z   z   x, y , z 
The Jacobian matrix is defined as
 x x x y x z 
 A  y x y y y z 
 z  x z  y z  z 
Lecture 16
13
Transforming Vector Functions
A vector function in two coordinate systems is related through the Jacobian matrix as follows.
 
E  r 
  A 
T
1
 
E r 
 

T 
E  r    A E   r  
Lecture 16
14
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Transforming Operations
An operation can also be transformed between two coordinate systems using the Jacobian matrix.

 A  F  r   A

 F   r    
det  A
T
 


T
1
 F  r    det  A   A  F   r     A
1
Lecture 16
15
Form Invariance
of Maxwell’s
Equations
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Maxwell’s Equations are Form
Invariant
In ANY coordinate system, Maxwell’s equations can be written as
Cartesian Coordinates


  H  j   E


  E   j    H
Spherical Coordinates
Cylindrical Coordinates


  H  j   E


  E   j    H


  H  j   E


  E   j    H
Martian Coordinates


  H  j   E


  E   j    H
We can transform Maxwell’s equations to a different coordinate system, but they still have the same form.


  H   j   E 


  E    j    H 
Lecture 16
17
Important Consequence
We can “absorb” the coordinate transformation completely into the material properties.


  H   j   E 


  E    j    H 


  H  j   E


  E   j    H
We are now back to the original coordinates, but the fields behave as if they are in the transformed coordinates.
Lecture 16
18
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Absorbing the Coordinate
Transformation into the Materials
Given the Jacobian [A] describing the coordinate transformation, the material property tensors are related through
   
 A   A
det  A
T
   
 A  A
det  A
 x x x y x z 
 A  y x y y y z 
 z  x z  y z  z 
T
Here we are actually transforming an operation, not a function.
We can think of this as a “stretching” matrix. It describes how much the coordinate changes in our transformed system with respect to a change in the original system.

 A  Jacobian transformation matrix from r

to r 
Lecture 16
19
Proof of Form Invariance (1 of 3)
We need to show that the following transform is true.


  E   j    H



  E    j    H 

 
r  r  r 
Defining the coordinate transformation as , we have
 
E  r  
T
1
 
E r 
T
1
 
H r 
  A 
 
H   r      A 

T
A    r    A


    r    
det  A
Lecture 16
20
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Proof of Form Invariance (2 of 3)
We substitute our transforms into the curl equation.


  E   j    H
 

T 
E  r    A E   r  




T
1
   r    det  A  A     r     A
This becomes

T 
T
1
   A E    j det  A  A     A





1
1

 

T 
H  r    A H   r  
 
T 
A H

 A      A
det  A
T


E    j    H 
Lecture 16
21
Proof of Form Invariance (3 of 3)
Recall the form of transforming an operation

T
A  F  r    A


 F   r    
det  A
We see that the group of terms around the curl operation indicates this is just the transformed curl.
 A     A E    j   H 
 
det  A


T


  E    j    H 

Lecture 16
22
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A Simple Example of the Proof
Start with Maxwell’s curl equation.
 
  
  E  r    j    r   H  r 
We define the following coordinate transform


r   ar
The terms transform according to
 
 
E  r   E  r


   r        r   
 
 
H  r   H   r

 0

 
  
z

 
 y



z
0

x
 

 0
y 



1 
     
x
a z


 
0 

 y




z 
0

x
 
y 

 
 
x

0 


The scale factor from the curl operation can be absorbed into the permeability.
 
  
  E  r    j  a   r   H  r 
Lecture 16
23
Transformation
Electromagnetics
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Concept of Transformation EM
Transformation electromagnetics is an analytical technique to calculate the permittivity and permeability functions that will bend fields in a prescribed manner.
Define a uniform grid with uniform rays.
Perform a coordinate transformation such that the rays follow some desired path.
Incorporate the coordinate transformation into the material tensors.
Lecture 16
25
Step 1: Pick a Coordinate
System
Pick a coordinate system that is most convenient for your device geometry.
Cartesian
Cylindrical
Spherical
Other?
Lecture 16
26
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Step 2: Draw Straight Rays
Through the Grid
Assuming we start with a wave in free space, draw the rays passing straight through the coordinate system.
x
y
z
1 0 0 
       0 1 0
0 0 1 
Lecture 16
27
Step 3: Define a Coordinate
Transform
Define a coordinate transform so that the rays will follow the desired path. Here, we are “squeezing” the wave at the center of the grid.
x  x

 x2 y 2 
y  y 1  exp   2  2  
 
 y  
x


z  z
Lecture 16
28
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Step 4: Calculate the Jacobian
Matrix
Given the coordinate transformation, the Jacobian matrix is calculated.
x
1
x
x
0
y
x
0
z
x  x

 x2 y 2 
y  y 1  exp   2  2  
 
 y  

x

z  z
1


x2 y2
 2 xy   x2   y2
 A   2 e
 x

0

 x2 y 2 
y 2 xy

exp   2  2 
 x y 
x  x2


0


0

1 
0
x2
y2
 2 y 2   2  2
1   2  1 e x y
 y



0
 2 y2 
 x2 y 2 
y
 1   2  1 exp   2  2 
 y

 x y 
y




y
0
z
z 
0
x
z 
0
y
z 
1
z
Lecture 16
29
Step 5: Calculate the Material
Tensors
The material tensors are calculated according to.
   
 A   A
det  A
T
   
 A  A
det  A
T
This is
 y2
 xx   xx   zz   zz 
 y2   2 y 2   y2  e
 xy   yx   zy   yx 

x2

y2
 x2  2y
2 xy y2






 x2  2 y 2   y2 1  e
x2

y2
 x2  2y




2
 2  x 2  y 2
x2 y 2
2 x2 2 y 2 

2 2 
 

 2 y e  x2  2y  1 e  x2  y2  1  4 x y e  x2  y2 
2
4
  2


y
x
 y


 xz   xz   yz   yz   zx   zx   zy   zy  0
x2
 yy   yy   y2 e
Lecture 16

y2
 x2  y2
30
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Plot of the Final Tensor Over
Entire Device
 xx   xx
 xy   xy
 xz   xz
 yx   yx
 yy   yy
 yz   yz
 zx   zx
 zy   zy
 zz   zz
Lecture 16
31
MATLAB Code to Generate the
Equations
x, y , z ,  x ,  y
% DEFINE VARIABLES
syms x y z;
syms sx sy;
1 0 0 
       0 1 0
% INITIALIZE MATERIALS
UR = eye(3,3);
ER = eye(3,3);
0 0 1 
x  x
% DEFINE COORDIANTE TRANSFORMATION
xp = x;
yp = y*(1 - exp(-x^2/sx^2)*exp(-y^2/sy^2));
zp = z;
% COMPUTE ELEMENTS OF JACOBIAN
A = [ diff(xp,x) diff(xp,y) diff(xp,z) ; ...
diff(yp,x) diff(yp,y) diff(yp,z) ; ...
diff(zp,x) diff(zp,y) diff(zp,z) ];
% ABSORB TRANSFORM INTO MATERIALS
UR = A*UR*A.'/det(A);
ER = A*ER*A.'/det(A);
% SHOW TENSOR
pretty(ER);
Lecture 16
   

 x2 y 2 
y  y 1  exp   2  2  
  x  y 



z  z
 x x x y x z 
 A  y x y y y z 
 z  x z y z  z 
 A   A
det  A
T
  
 A  A
det  A
T
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MATLAB Code to Fill Grid
% PARAMETERS
ER = subs(ER,sx,0.25);
ER = subs(ER,sy,0.25);
% FILL GRID
for ny = 1 :
for nx =
er =
er =
Ny
1 : Nx
subs(ER,x,X(nx,ny));
subs(er,y,Y(nx,ny));
ERxx(nx,ny) = er(1,1);
ERxy(nx,ny) = er(1,2);
ERxz(nx,ny) = er(1,3);
ERyx(nx,ny) = er(2,1);
ERyy(nx,ny) = er(2,2);
ERyz(nx,ny) = er(2,3);
ERzx(nx,ny) = er(3,1);
ERzy(nx,ny) = er(3,2);
ERzz(nx,ny) = er(3,3);
This is VERY slow!!!
Recommend calculating analytical equations and then manually typing these into MATLAB to construct the device.
end
end
Lecture 16
33
Stretching Space
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Setup of the Problem
Starting coordinate system
We want to “stretch” space to move two objects farther apart.
We calculate metamaterial properties that will do this.
Lecture 16
Slide 35
Define the Coordinate Transform
Suppose we want to “stretch” the z‐axis by a factor of a.
x  x
y  y
z  z a
Lecture 16
36
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Calculate the Jacobian
x  x
y  y
 x x x y x z  1 0 0 
 A  y x y y y z   0 1 0 
 z  x z  y z  z   0 0 1 a 
z  z a
Lecture 16
37
Calculate  and 
   
 A   A
det  A
T
1 0 0   
0 1 0   0


 0 0 a   0


0

1
det 0
0
0
T
0  1 0 0 
0   0 1 0 
   0 0 1 a 

0 0
1 0 
0 1 a 
T
   
Lecture 16
 A  A
det  A
T
1 0 0   0 0  1 0 0 
0 1 0   0  0  0 1 0 




0 0 a   0 0   0 0 1 a 


1 0 0 
det 0 1 0 
0 0 1 a 

0

 0
0

0
1a
0 
0 
 a 2 
 a
  0
 0
0 
 0
0 
0 

 a
 0 0  a 2  
0
1a
 0
0
a
0
0
a
0
0 
0 
 a 
0 
0 
 a 
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What Does the Answer Mean?
 a
     0
 0
0
a
0
 a
    0
 0
0 
0 
 a 
0
a
0
0 
0 
 a 
These tensors have two equal elements and a different third element.  Uniaxial
For a > 0, the third element is smaller in value than the first two.  Negative uniaxial
Lecture 16
39
How Do We Realize This
Metamaterial?
Recall from Lecture 13, a negative uniaxial metamaterial is an array of sheets.
 o
    0
 0
 o  a
0
o
0
0
0 
 e 
High 
y
e   a
Stretching factor given tensor:
a  o e
x High 
z
Low 
e  o
Negative Uniaxial
   o e
Lecture 16
40
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Cloaking
What is Cloaking?
free space
object
cloak
A true cloak must have the following properties:
1. Cannot reflect or scatter waves.
2. Must perfectly reconstruct the wave front on the other side of the object.
3. Must work for waves applied from any direction.
Lecture 16
D.‐H. Kwon, D. H. Werner, “Transformation Electromagnetics: An Overview of the Theory and Applications,” IEEE Ant. Prop. Mag., Vol. 52, No. 1, 24‐46 (2020).
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Famous Cylinder Cloak
Lecture 16
43
Path of Rays for Invisibility
To render a region in the center of the grid invisible, the wave must be made to flow around it.
Lecture 16
44
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Coordinate Transformation
Geometry
Lecture 16
45
Deriving the Coordinate
Transformation
Be observation, we wish to map our coordinates as follows.
     0   R1
     R2   R2
This transform can be done with a simple straight line.
y  mx  b
    m  b
y -intercept: b  R1
Slope: m 
Lecture 16
R2  R1
R2
 
R2  R1
  R1
R2
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The Jacobian Matrix
Due to the cylindrical geometry, the coordinate transformation will be in cylindrical coordinates.
   R1 
R2  R1

R2
  
z  z
It follows that the Jacobian matrix is
  
 

   
 A  
 
 z 
 

 

  

z 

  
z 
 R2  R1  R2
    
0

z  

0
z   
z 
0
0

   0
0
1 
Lecture 16
47
The Material Tensors
The material tensors are

T
A    r    A


    r '  
det  A

T
A   r    A


   r '  
det  A
Assuming we start with free space, we will have

 r   R1
 r



    r '      r '     0



 0

Lecture 16
0
0
r
r   R1
0
0
 R2 


 R2  R1 
2







r   R1 

r 
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Cylindrical Tensors


rr  r '   rr  r '


  r '    r '


 zz  r '   zz  r '
0
∞
0
r   R1
r
r
r   R1
 R2  r   R1


r
 R2  R1 
R1  0.10
2
R2  0.45
Lecture 16
49
Physical Device


   r '     r '
Duke University
Lecture 16
50
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Cartesian Tensors
Accuracy of these results is not guaranteed!
Lecture 16
51
Carpet
Cloaking
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Problems with Cloaking by
Transformation Electromagnetics
• The resulting materials:
– Are anisotropic
– Require dielectric and magnetic properties
– Require extreme and singular values
• Particularly problematic at optical frequencies
Lecture 16
53
Three Cases for Cloaking
Squish an object to a sheet.
Squish an object to a point.
Squish an object to a line.
Object becomes infinitely conducting.
This is not a problem because the objects have zero size.
These can be rendered invisible, but require extreme and singular values as well as being anisotropic.
Lecture 16
A sheet is highly visible.
It can only be made invisible if it sits on another conducting sheet so they cannot be distinguished.
While more limited, invisibility can be realized without extreme values and with isotropic materials.
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Carpet Cloak Concept
J. Li, J. B. Pendry, “Hiding under the Carpet: A New Strategy for Cloaking,” Phys. Rev. Lett. 101, 203901 (2008).
Lecture 16
55
The Jacobian Matrix and
Covariant Matrix
We define the Jacobian matrix [A] just like we did before.
Aij 
xi
xi
The Covariant matrix is then
 g    A  A
T
Lecture 16
56
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The Original System
 ref
ref  1
Lecture 16
57
The Transformed System

Lecture 16
 
58
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The Transformation

 ref
det  g 
 A A
  
det  g 
T
 ref anything
ref  1
Lecture 16
59
A Preliminary Tensor Description
The permeability is still a tensor quantity. For a given wave, it will have two principle values.
    T
and  L
T  L  1
This gives us two refractive indices to characterize the medium.
nT  T 
nL   L
We can characterize the anisotropy using the anisotropy factor .
 nT nL 
,

n
 L nT 
  max 
Lecture 16
60
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Isotropic Medium
By picking a suitable transform, we can neglect the permeability.

 ref
det  g 
1 0 0 
    0 1 0
0 0 1 
The optimal transformation is generated by minimizing the Modified‐Liao functional
h  Tr  g  
1 w


dx
dy


0
0
det  g 
hw
2
This leads to high  and low anisotropy.
Lecture 16
61
Impact of Grid on Isotropic
Approximation
transfinite grid
Low  so this medium will be poorly approximated as isotropic.
High  so this medium is well approximated as isotropic.
Lecture 16
quasiconformal grid

 ref
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Example Carpet Cloak
Scattering from cloaked object.
Scattering from just the object.
Lecture 16
63
Other
Applications
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Electromagnetic Cloaking of
Antennas (1 of 2)
An antenna can be cloaked to either render it invisible to a “bad guy” or to reduce the effects of scattering and coupling to nearby objects.
This implies a “dual band” mode of operation because the cloak must be transparent to the radiation frequency of the antenna.
Metamaterials used to realize cloaks are inherently narrowband which is an advantage for this application.
Lecture 16
65
Electromagnetic Cloaking of
Antennas (2 of 2)
Antenna A1 transmits at frequency f1.
Antenna A2 transmits at frequency f2.
f1
A1
f2
A2
A1 transmitting through uncloaked A2
Lecture 16
Cloak of A1 must be transparent to f1.
Cloak of A2 must be transparent to f2.
A1 transmitting through cloaked A2
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Reshaping the Scattering of
Objects
Scattering of an elliptical shaped object.
Rectangular object embedded in an anisotropic medium designed by TEM to scatter like the elliptical object.
O. Ozgun, M. Kuzuoglu, “Form Invariance of Maxwell’s Equations: The Pathway to Novel Metamaterial Specifications for Electromagnetic Reshaping,” IEEE Ant. And Prop. Mag., Vol. 52, No. 3, 51‐65 (2010).
Lecture 16
67
Polarization Splitter
Device is made uniform in the z direction.
Maxwell’s equations split into two independent modes.
Each mode can be independently controlled.
E Mode
Lecture 16
H Mode
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Polarization Rotator
Lecture 16
69
Wave Collimator
Lecture 16
70
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Flat Lenses
Lecture 16
71
Beam Benders
Lecture 16
72
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Concluding
Remarks
Notes
• Transformation electromagnetics (TEM) is a coordinate
transformation technique where the transformation is
“absorbed” into the permittivity and permeability
functions.
• The method produces the permittivity and permeability as
a function of position.
• The method does not say anything about how the materials
will be realized.
• The resulting material functions will be anisotropic and
require dielectric and magnetic metamaterials with often
extreme values. This leads to structures requiring metals.
• TEM is perhaps more commonly called transformation
optics (TO) because that is where it first appeared.
Lecture 16
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37