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 4 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 5 Design Process Using Spatial Transforms Step 3 of 4: Map Properties to Engineered Materials & 6 3 1/29/2020 Design Process Using Spatial Transforms Step 4 of 4: Generate Overall Lattice 7 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 9 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. 10 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) 11 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 rfsin 0 J yr y f yz r f z zr z f zz y y y sinf rfcos 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 12 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 cosqr sin q 0 13 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 14 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 15 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 16 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 18 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. 19 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 20 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 detJ 21 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 11 1/29/2020 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 23 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 24 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. 26 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? 27 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 28 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 29 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 30 15 1/29/2020 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 32 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) ; ..