Transformation Optics

1/29/2020

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

Slide 2

1 1/29/2020

Transformation Optics

Slide 3

Design Process Using Spatial Transforms

Step 1 of 4: Define Spatial Transform

2 1/29/2020

Design Process Using Spatial Transforms

Step 2 of 4: Calculate Effective Material Properties

    E  jH E  j H     H  j E H  j E

Design Process Using Spatial Transforms

Step 3 of 4: Map Properties to Engineered Materials

 & 

3 1/29/2020

Design Process Using Spatial Transforms

Step 4 of 4: Generate Overall Lattice

Coordinate Transformation

Slide 8

4 1/29/2020

Coordinate Transformation

One coordinate space 푟⃗ can be mapped to another 푟⃗.

r  xxˆ  yy ˆ  zzˆ r xx ˆ  yy  ˆ  zz ˆ

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.

5 1/29/2020

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)

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 

6 1/29/2020

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 cosqr sin q 0 

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  

7 1/29/2020

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    

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         detJ 

1 1  T Fr  det JJFr        J  

8 1/29/2020

Form Invariance of Maxwell’s Equations

Slide 17

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 

9 1/29/2020

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.

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 detJ  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 

10 1/29/2020

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         detJ 

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  jdet JJ 1    J TT  JH     J   J T E j    H  detJ  22

11 1/29/2020

Proof of Form Invariance (3 of 3)

Recall the form of transforming an operation J Fr   J T F r         detJ 

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 detJ   

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      

12 1/29/2020

Analytical Transformation Optics

Slide 25

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.

13 1/29/2020

Step 1: Pick a Coordinate System

Pick a coordinate system that is most convenient for your device geometry.

Cartesian Cylindrical Spherical

Other?

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 

14 1/29/2020

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

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  2xyxy 2 y  xy x Je  1  1  e 0  z 2  2   0 x y   y z 0 0 1   1   z

15 1/29/2020

Step 5: Calculate the Material Tensors

The material tensors are calculated according to. JJ T JJ    T         detJ  det J  The elements of these tensors are  2        y xx xx zz zz x2 y 2 2  2 2 2 2 x  y y2y   y  e 2 2xy y xy  yx   zy   yx  x2 y 2   2 2 22y 2  2  1  ex  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    2eeexy  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

16 1/29/2020

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 detJ  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

17 1/29/2020

Stretching Space

Slide 35

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.

Slide 36

18 1/29/2020

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

19 1/29/2020

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  detJ  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  detJ  1 0 0  1 a     0 0  a  det 0 1 0  0 0 1 a 

What Does the Answer Mean?

a0 0   a  0 0         0a  0       0 a 0  0 0a   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

20 1/29/2020

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

Cloaking

Slide 42

21 1/29/2020

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

22 1/29/2020

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.

Coordinate Transformation Geometry

23 1/29/2020

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

1r 1  r  1  r   1r rf  1 z    R2 R 1  R 2 0 0  rf rf   rf      J    0r r 0 1r rf  1 z      0 0 1  1z 1  z  1  z       1r rf  1 z  48

24 1/29/2020

The Material Tensors

The material tensors are

J   rJ   T J   rJ   T r '      r '        detJ    detJ 

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 

Cylindrical Tensors

  r'   r ' r'   r ' rrr'   rr  r ' qq  qq   zz  zz  

0 ∞ 0

r 2 r  R1 R2  rR  1  r  R   r 1 RR2 1  r

R10.10 R 2  0.45

25 1/29/2020

Physical Device

r'    r '

Duke University

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

26 1/29/2020

Carpet Cloaking

Slide 53

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

27 1/29/2020

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.

Carpet Cloak Concept

J. Li, J. B. Pendry, “Hiding under the Carpet: A New Strategy for Cloaking,” Phys. Rev. Lett. 101, 203901 (2008). 56

28 1/29/2020

The Jacobian Matrix and Covariant Matrix

Define the Jacobian matrix [J] just like before.

xi Aij  xi

The Covariant matrix 푔 is then

g   JT  J 

The Original System

 ref ref 1

29 1/29/2020

The Transformed System

 

The Transformation

   ref detg   ref anything T J  J  ref 1    detg 

30 1/29/2020

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 

Isotropic Medium

By picking a suitable transform, the permeability can be neglected. 1 0 0   ref      0 1 0  detg    0 0 1 

The optimal transformation is generated by minimizing the Modified-Liao functional

2 1 w h Trg   dx dy  This leads to high  and low anisotropy. hw0  0 det g 

31 1/29/2020

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.

Example Carpet Cloak

Scattering from cloaked object.

Scattering from just the object.

32 1/29/2020

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.

33 1/29/2020

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

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

34 1/29/2020

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

Polarization Rotator

35 1/29/2020

Wave Collimator

Flat Lenses

36 1/29/2020

Beam Benders

Concluding Remarks

Slide 74

37 1/29/2020

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).

Conformal Mapping

Slide 76

38 1/29/2020

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?

Complex Numbers as Coordinates

Imu

Reu

39 1/29/2020

Coordinate Transform with Complex Numbers

 u

v L u

 v

Observe the constant orthogonality of the coordinate axes.  Conformal mapping