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Advanced Electromagnetics: 21st Century Electromagnetics Transformation Optics
Lecture Outline
• Transformation Optics • Coordinate transformations • Form invariance of Maxwell’s equations • Analytical transformation optics • Stretching space • Cloaking • Carpet cloaking • Other applications • Concluding remarks • Conformal Mapping
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Transformation Optics
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Design Process Using Spatial Transforms
Step 1 of 4: Define Spatial Transform
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Design Process Using Spatial Transforms
Step 2 of 4: Calculate Effective Material Properties
E jH E j H H j E H j E
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Design Process Using Spatial Transforms
Step 3 of 4: Map Properties to Engineered Materials
&
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Design Process Using Spatial Transforms
Step 4 of 4: Generate Overall Lattice
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Coordinate Transformation
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Coordinate Transformation
One coordinate space 푟⃗ can be mapped to another 푟⃗ .
r xxˆ yy ˆ zzˆ r xx ˆ yy ˆ zz ˆ
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Jacobian Matrix [J]
To aid in the coordinate transform, the Jacobian transformation matrix is used.
x x x x y z T y y y J r Each term quantifies the “stretching” of the coordinates. x y z z z z Gradient of a vector is a tensor! Wow! x y z
The Jacobian matrix does not perform a coordinate transformation. It transforms functions and operations between different coordinate systems.
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General Form of the Jacobian Matrix [J]
hx hx hx 2 1 1 1 1 1 1 n x h k hx hx hx i 1 1 2 2 3 3 k 1 x 'i hx2 2 hx 2 2 hx 2 2 J Coordinate h1 h2 h3 hx1 1 hx 2 2 hx 3 3 System Cartesian 1 1 1 (x,y,z) hx3 3 hx 3 3 hx 3 3 Cylindrical hx hx hx 1 r 1 1 1 2 2 3 3 (r,f,z) Spherical 1 r rsinq (r,f,q)
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Example #1: Cylindrical to Cartesian The Cartesian and cylindrical coordinates are related through x rcos f y rsin f z z The Jacobian matrix is then
xr x f xz x x x cosf rf sin 0 J yr y f yz r f z zr z f zz y y y sinf rf cos 0 r f z cosf r sin f 0 z z z J sinf r cos f 0 0 0 1 r f z 0 0 1
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Example #2: Spherical to Cartesian The Cartesian and spherical coordinates are related through x r sinq cos f y r sinq sin f z r cosq The Jacobian matrix is then xrx q x f x x x J yry q y f sinqf cos r cos qf cos r sin qf sin r q f zrz q z f y y y sinsinqf r cossin qf r sincos qf r q f sinqf cosr cos qf cos r sin qf sin z z z cosq r sin q 0 J sinqf sin r cos qf sin r sin qf cos r q f cosqr sin q 0
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Example #3: Cartesian to Cartesian
Coordinates in Cartesian space can be transformed according to x xxyz ,, y yxyz ,, z zxyz ,, The Jacobian matrix is defined as xx xy xz J yxyyyz zx zy zz
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Transforming Vector Functions
The same vector function (variable that changes as a function of position) expressed in two different coordinate systems is related through the Jacobian matrix 퐽 as follows.
1 Er J T Er
Er JEr T
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Transforming Operations
An operation (think derivatives, integrals, tensors, etc.) can also be transformed between two coordinate systems using the Jacobian matrix 퐽 .
J Fr J T F r detJ
1 1 T Fr det JJFr J
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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 Spherical Coordinates Hj E Cylindrical Coordinates Hj E Martian Coordinates E j H Hj E E j H Hj E E j H E j H
Maxwell’s equations can be transformed to a different coordinate system, but they still have the same form.
Hj E E j H
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Important Consequence
The math associated with the coordinate transformation can be “absorbed” completely into the material properties.
Hj E Hj E E j H Ej H
Maxwell’s equations are now back to the original coordinates, but the fields behave as if they are in the transformed coordinates.
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Absorbing the Coordinate Transformation into the Materials Given the Jacobian [J] describing the coordinate transformation, the material property tensors are related through
JJ T JJ T Here, an operation is being detJ det J transformed, not a function.
xx xy xz Think of this as a matrix that characterizes the “stretching” of the J yxyyyz transform. It describes how much the coordinate changes in the transformed system with respect to a change in the original system. zx zy zz
J Jacobian transformation matrix from rr to
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Proof of Form Invariance (1 of 3)
It is needed to show that the following transform is true.
EjH EjH
Defining the coordinate transformation as r rr , the functions can be transformed as
T 1 Er J Er T 1 Hr J Hr J rJ T r detJ
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Proof of Form Invariance (2 of 3)
Substitute the transforms into the original curl equation. E j H
T T Er JEr Hr JHr 1 T 1 r det JJ rJ
This becomes 1 JE T jdet JJ 1 J TT JH J J T E j H detJ 22
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Proof of Form Invariance (3 of 3)
Recall the form of transforming an operation J Fr J T F r detJ
The group of terms around the curl operation indicates this is just the transformed curl operation.
T J J E j H E j H detJ
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A Simple Example of the Proof Start with Maxwell’s curl equation. Er j rHr We define the following coordinate transform r ar The terms transform according to 0 0 Er Er zy zy 1 r r 0 0 z x az x 0 0 Hr Hr yx yx The scale factor from the curl operation can be absorbed into the permeability. Er jarHr
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Analytical Transformation Optics
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Concept of Transformation EM
Transformation optics is an analytical technique to calculate the permittivity and permeability functions that will bend fields in a prescribed manner.
Define a uniform Perform a coordinate Move the coordinate grid with uniform transformation such transformation into the rays. that the rays follow material tensors. some desired path.
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Step 1: Pick a Coordinate System
Pick a coordinate system that is most convenient for your device geometry.
Cartesian Cylindrical Spherical
Other?
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Step 2: Draw Straight Rays Through the Grid
Start with a plane 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
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Step 3: Define a Coordinate Transform
Define a coordinate transform so that the rays will follow the desired path. Here, the wave is being “squeezed” at the center of the grid.
x x x2 y 2 y y 1 exp 2 2 x y z z
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Step 4: Calculate the Jacobian Matrix
Given the coordinate transformation, the Jacobian matrix 퐽 is calculated.
x 1 x x x x 0 y 2 2 x x y 0 y y 1 exp z 2 2 x y y 2 xy x2 y 2 exp x 2 2 2 z z x xy y 2 y2 xy 2 2 1 1 exp 2 2 2 y y xy y 1 0 0 0 z xy2 2 xy 2 2 z 2 0 2 2 2 2 2xyxy 2 y xy x Je 1 1 e 0 z 2 2 0 x y y z 0 0 1 1 z
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Step 5: Calculate the Material Tensors
The material tensors are calculated according to. JJ T JJ T detJ det J The elements of these tensors are 2 y xx xx zz zz x2 y 2 2 2 2 2 2 x y y2y y e 2 2xy y xy yx zy yx x2 y 2 2 2 22y 2 2 1 ex y x y 2 xy2 2 xy 2 2 xy 2 22 xy 2 2 2 2 2 2 2 22y 2 2 1 2 2 4 xy 2 2 2eeexy xy xy 1 e xy yy yy y 2 2 4 y y x
xz xz yz yz zx zx zy zy 0 31
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
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MATLAB Code to Generate the Equations (Analytical Approach)
% DEFINE VARIABLES x, y , z , x , y syms x y z; syms sx sy; 1 0 0 % INITIALIZE MATERIALS 0 1 0 UR = eye(3,3); ER = eye(3,3); 0 0 1
% DEFINE COORDIANTE TRANSFORMATION x x xp = x; yp = y*(1 - exp(-x^2/sx^2)*exp(-y^2/sy^2)); x2 y 2 y y 1 exp zp = z; 2 2 x y % COMPUTE ELEMENTS OF JACOBIAN z z J = [ diff(xp,x) diff(xp,y) diff(xp,z) ; ... diff(yp,x) diff(yp,y) diff(yp,z) ; ... xx xy xz diff(zp,x) diff(zp,y) diff(zp,z) ]; J yxyyyz % ABSORB TRANSFORM INTO MATERIALS zx zy zz UR = J*UR*J.'/det(J); ER = J*ER*J.'/det(J); JJ T JJ T % SHOW TENSOR detJ det J pretty(ER); 33
MATLAB Code to Fill Grid (Analytical Approach)
% PARAMETERS ER = subs(ER,sx,0.25); ER = subs(ER,sy,0.25); X and Y generated by MATLAB’s meshgrid() command.
% FILL GRID for ny = 1 : Ny for nx = 1 : Nx er = subs(ER,x,X(nx,ny)); er = subs(er,y,Y(nx,ny));
ERxx(nx,ny) = er(1,1); ERxy(nx,ny) = er(1,2); This is VERY slow!!! ERxz(nx,ny) = er(1,3);
ERyx(nx,ny) = er(2,1); Recommend calculating analytical equations ERyy(nx,ny) = er(2,2); and then manually typing these into MATLAB ERyz(nx,ny) = er(2,3); to construct the device. ERzx(nx,ny) = er(3,1); ERzy(nx,ny) = er(3,2); ERzz(nx,ny) = er(3,3); end end 34
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Stretching Space
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Setup of the Problem
Starting We want to “stretch” We calculate coordinate system space in order to move metamaterial properties two objects farther that will do this. apart.
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Define the Coordinate Transform Suppose it is desired to “stretch” the z-axis by a factor of a.
It is desired to end in the standard uniform Cartesian coordinate system. x x This requires starting in a coordinate system that is stretched. Therefore, the coordinate transform must compress space. This will give [] and [] y y that stretch space. z z a
xyz,, x,, y z 37
Calculate the Jacobian
x x xx xy xz 1 0 0 y y J yxyyyz 0 1 0 z z a z x z y z z 0 0 1 a
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Calculate Transformed and
T 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 T a 0 0 J J 0 0 a 0 0 0 0 1 a 0 0 a2 0a 0 detJ 1 0 0 1 a 0 0 a det 0 1 0 0 0 1 a
T 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 T a 0 0 J J 0 0 a 0 0 0 0 1 a 0 0 a2 0a 0 detJ 1 0 0 1 a 0 0 a det 0 1 0 0 0 1 a
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What Does the Answer Mean?
a0 0 a 0 0 0a 0 0 a 0 0 0a 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
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How Do We Realize This Metamaterial?
Recall from Subwavelength Gratings lecture, a negative uniaxial metamaterial is realized by an array of sheets with alternating material properties.
o 0 0 0 0 o 0 0 e High y o a
e a x High z Stretching factor given tensor: Low a o e e o Negative Uniaxial
o e
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Cloaking
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What is Cloaking?
free space object cloak
A perfect 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.
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). 43
Famous Cylinder Cloak
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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.
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Coordinate Transformation Geometry
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Deriving the Coordinate Transformation
Be observation, we wish to map our coordinates as follows.
r r 0 R1
r r R2 R 2
This transform can be done with a simple straight line.
y mx b r m r b
R2 R 1 y-intercept: bR 1 r r R R 1 R R 2 Slope: m 2 1 R 2 47
The Jacobian Matrix
Due to the cylindrical geometry, the coordinate transformation will be in cylindrical coordinates.
R2 R 1 r R1 r R2 f f z z
It follows that the Jacobian matrix is
1r 1 r 1 r 1r rf 1 z R2 R 1 R 2 0 0 rf rf rf J 0r r 0 1r rf 1 z 0 0 1 1z 1 z 1 z 1r rf 1 z 48
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The Material Tensors
The material tensors are
J rJ T J rJ T r ' r ' detJ detJ
Assuming we start with free space, we will have
r R 1 0 0 r r r' r ' 0 0 r R1 2 R rR 0 0 2 1 RR2 1 r
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Cylindrical Tensors
r' r ' r' r ' rrr' rr r ' qq qq zz zz
0 ∞ 0
r 2 r R1 R2 rR 1 r R r 1 RR2 1 r
R10.10 R 2 0.45
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Physical Device
r' r '
Duke University
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Cartesian Tensors
xx xy xz 5 2 3
0 -2 -3 yx yy yz 2 5 3
-2 0 -3 zx zy zz 3 3 1.5
-3 -3 0
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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
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Three Cases for Cloaking
Squish an object to a sheet. Squish an object Squish an object to a point. to a line.
Object becomes infinitely conducting. A sheet is highly visible. This is not a problem because the objects have zero It can only be made invisible if it sits on size. another conducting sheet so they cannot be distinguished. These can be rendered invisible, but require extreme and singular values as well as being anisotropic. 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). 56
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The Jacobian Matrix and Covariant Matrix
Define the Jacobian matrix [J] just like before.
xi Aij xi
The Covariant matrix 푔 is then
g JT J
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The Original System
ref ref 1
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The Transformed System
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The Transformation
ref detg ref anything T J J ref 1 detg
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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 two refractive indices to characterize the medium.
nTT n LL
The anisotropy can be characterized using the anisotropy factor a.
nT n L a max , nL n T
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Isotropic Medium
By picking a suitable transform, the permeability can be neglected. 1 0 0 ref 0 1 0 detg 0 0 1
The optimal transformation is generated by minimizing the Modified-Liao functional
2 1 w h Trg dx dy This leads to high and low anisotropy. hw0 0 det g
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Impact of Grid on Isotropic Approximation
transfinite grid
Low so this medium will be poorly approximated as isotropic. quasiconformal grid ref High so this medium is well approximated as isotropic.
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Example Carpet Cloak
Scattering from cloaked object.
Scattering from just the object.
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Other Applications
Slide 65
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.
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Electromagnetic Cloaking of Antennas (2 of 2)
Antenna A1 transmits at frequency f1. Antenna A2 transmits at frequency f2.
f1 A f2 A2 1 Cloak of A1 must be transparent to f1. Cloak of A2 must be transparent to f2.
A1 transmitting through uncloaked A2 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). 68
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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 H Mode
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Polarization Rotator
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Wave Collimator
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Flat Lenses
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Beam Benders
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Concluding Remarks
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Notes
• Transformation optics (TO) is a coordinate transformation technique where the transformation is “absorbed” into the permittivity and permeability functions. • The method applies to any frequency, not just optical frequencies. • 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. • TO is also sometimes called transformation electromagnetics (TEM).
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Conformal Mapping
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Observations About TO
Observation #1 Observation #2 Sharper bends lead to more Oblique coordinates lead to extreme material properties. anisotropic material properties.
Least extreme
Isotropic Weakly Strongly Most anisotropic Anisotropic extreme
How to we always enforce this?
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Complex Numbers as Coordinates
Imu
Reu
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Coordinate Transform with Complex Numbers
u
v L u
v
Observe the constant orthogonality of the coordinate axes. Conformal mapping
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