Lecture #16. Spatial Transforms. Lecture 16 1

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1 ECE 53 1 st Century Electromagnetics Instructor: Office: Phone: E Mail: Dr. Raymond C. Rumpf A 337 (915) rcrumpf@utep.edu Lecture #16 Spatial ransforms Lecture 16 1 Lecture Outline ransformation Optics Coordinate transformations Form invariance of Maxwell s equations ransformation electromagnetics Stretching space Cloaking Carpet cloaking Other applications Concluding remarks Conformal Mapping Lecture 16 Slide 1

2 ransformation Optics Design Process Using Spatial ransforms Step 1 of 4: Define Spatial ransform Lecture 16 4

3 Design Process Using Spatial ransforms Step of 4: Calculate Effective Material Properties E jh H j E E j H H j E Lecture 16 5 Design Process Using Spatial ransforms Step 3 of 4: Map Properties to Engineered Materials & Lecture

4 Design Process Using Spatial ransforms Step 4 of 4: Generate Overall Lattice Lecture 16 7 Coordinate ransformation 4

5 Coordinate ransformation We can map one coordinate space into another r xxˆ yyˆzzˆ r xx ˆ yy ˆzz Lecture 16 9 Jacobian Matrix [J] o aid in the coordinate transform, we use the Jacobian transformation matrix. x x x x y z y y y x y z z z z x y z J r Gradient of a vector is a tensor! Wow! Each term quantifies the stretching of the coordinates. he Jacobian matrix does not perform a coordinate transformation. It transforms functions and operations between different coordinate systems. Lecture

6 General Form of the Jacobian Matrix [J] J h 1 x 1 h 1 x 1 h 1 x 1 h1 x1 h x h3 x3 h x h x h x h1 x1 h x h3 x3 h 3 x 3 h 3 x 3 h 3 x 3 h1 x1 h x h3 x3 n x k hi k 1 x ' i Coordinate System Cartesian (x,y,z) Cylindrical (,,z) Spherical (r,,) h 1 h h r rsin Lecture Example #1: Cylindrical to Cartesian he Cartesian and cylindrical coordinates are related through x cos y sin z z he Jacobian matrix is then x x x z J y y y z z z z z J cos sin 0 sin cos x x x cos sin 0 z y y y sin cos 0 z z z z z Lecture

7 Example #: Spherical to Cartesian he Cartesian and spherical coordinates are related through x rsin cos y rsin sin z rcos he Jacobian matrix is then x r x x A y r y y z r z z sin cos rcoscos rsinsin A sin sin rcossin rsincos cos r sin 0 x x x sin cos rcoscos rsinsin r y y y sin sin rcossin rsin cos r z z z cos r sin 0 r Lecture 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 he Jacobian matrix is defined as x x x y x z J y x y y y z z x z y z z Lecture

8 ransforming Vector Functions A vector function (variable that changes as a function of position) in two coordinate systems is related through the Jacobian matrix as follows. 1 E r J E r E r J E r Lecture ransforming Operations An operation (think derivatives, integrals, tensors, etc.) can also be transformed between two coordinate systems using the Jacobian matrix. F r J F r J det J F r J J F r J 1 1 det Lecture

9 Form Invariance of Maxwell s Equations 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 Cylindrical Coordinates H j E E j H Spherical Coordinates 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. Hj E Ej H Lecture

10 Important Consequence We can absorb the coordinate transformation completely into the material properties. Hj E Ej H H j E E j H We are now back to the original coordinates, but the fields behave almost as if they are in the transformed coordinates. Lecture Absorbing the Coordinate ransformation into the Materials Given the Jacobian [J] describing the coordinate transformation, the material property tensors are related through J J J J J det detj Here we are actually transforming an operation, not a function. x x x y x z J y x y y y z z x z y z z 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. J r r Jacobian transformation matrix from to Lecture

11 Proof of Form Invariance (1 of 3) We need to show that the following transform is true. E j H Ej H Defining the coordinate transformation as r r r, we have 1 E r J E r 1 H r J H r r J r J det J Lecture 16 1 Proof of Form Invariance ( of 3) We substitute our transforms into the curl equation. E j H E r J E r 1 r detj J r J 1 H r J H r his becomes 1 det J det J J 1 J E j J J J J H Ej H Lecture 16 11

12 Proof of Form Invariance (3 of 3) Recall the form of transforming an operation J FrJ F r detj We see that the group of terms around the curl operation indicates this is just the transformed curl. J J J EjH det Ej H Lecture 16 3 A Simple Example of the Proof Start with Maxwell s curl equation. Er j r Hr We define the following coordinate transform r ar he terms transform according to Er Er r r Hr Hr 0 0 z y z y z x a z x 0 0 y x y x he scale factor from the curl operation can be absorbed into the permeability. Er j ar Hr Lecture

13 Analytical ransformation Electromagnetics Concept of ransformation EM ransformation 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. Move the coordinate transformation into the material tensors. Lecture

14 Step 1: Pick a Coordinate System Pick a coordinate system that is most convenient for your device geometry. Cartesian Cylindrical Spherical Other? Lecture 16 7 Step : Draw Straight Rays hrough the Grid Assuming we start with a wave in free space, draw the rays passing straight through the coordinate system. x y z Lecture

15 Step 3: Define a Coordinate ransform 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 x y y y1exp x y z z Lecture 16 9 Step 4: Calculate the Jacobian Matrix Given the coordinate transformation, the Jacobian matrix is calculated. x x x y y y1exp x y z z x y x y xy x y y x y J e 1 1 e 0 x y x 1 x x 0 y x 0 z y xy x y exp x x x y y y x y 1 1 exp y y x y y 0 z z 0 x z 0 y z 1 z Lecture

16 Step 5: Calculate the Material ensors he material tensors are calculated according to. his is J J J J J det detj xx xx zz zz x y x y y y y e xy y xy yx zy yx x y x y x y y 1e 0 y x y x y x y x y 1 4 x y y x y x y x y x y yy yy ye e e 1 e 4 y y x xz xz yz yz zx zx zy zy Lecture Plot of the Final ensor Over Entire Device xx xx xy xy xz xz yx yx yy yy yz yz zx zx zy zy zz zz Lecture

17 MALAB Code to Generate the Equations (Analytical Approach) % DEFINE VARIABLES syms x y z; syms sx sy; % INIIALIZE MAERIALS UR = eye(3,3); ER = eye(3,3); x, y, z,, x y % DEFINE COORDIANE RANSFORMAION xp = x; yp = y*(1 - exp(-x^/sx^)*exp(-y^/sy^)); zp = z; % COMPUE ELEMENS OF JACOBIAN J = [ 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 RANSFORM INO MAERIALS UR = J*UR*J.'/det(J); ER = J*ER*J.'/det(J); % SHOW ENSOR pretty(er); x x x y y y1exp z z x y Lecture x x x y x z J y x y y y z z x z y z z J J J J J det detj MALAB Code to Fill Grid (Analytical Approach) % PARAMEERS ER = subs(er,sx,0.5); ER = subs(er,sy,0.5); % 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,); ERxz(nx,ny) = er(1,3); ERyx(nx,ny) = er(,1); ERyy(nx,ny) = er(,); ERyz(nx,ny) = er(,3); ERzx(nx,ny) = er(3,1); ERzy(nx,ny) = er(3,); ERzz(nx,ny) = er(3,3); end end X and Y generated by MALAB s meshgrid() command. his is VERY slow!!! Recommend calculating analytical equations and then manually typing these into MALAB to construct the device. Lecture

18 Stretching Space Setup of the Problem Starting coordinate system We want to stretch space in order to move two objects farther apart. We calculate metamaterial properties that will do this. Lecture 16 Slide 36 18

19 Define the Coordinate ransform Suppose we want to stretch the z axis by a factor of a. x x y y z z a We wish to end in the standard uniform Cartesian coordinate system. his means we must start in a coordinate system that is stretched. herefore, the coordinate transform must compress space. his will give [] and [] that stretch space. x, yz, x, y, z Lecture Calculate the Jacobian x y z x y z a x x x y x z J y x y y y z z x z y z z a Lecture

20 Calculate and a JJ 0 0 a a 0 0 a 0 a 0 detj a 0 0 a det a a 0 0 J J 0 0 a a 0 0 a 0 a a 0 0 a det a detj Lecture What Does the Answer Mean? a 0 0 a a a 0 0 a a hese 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

21 How Do We Realize his Metamaterial? Recall from Lecture 13, a negative uniaxial metamaterial is an array of sheets. o o e a o a e Stretching factor given tensor: a o o e e High y x High z Low e o Negative Uniaxial Lecture Cloaking 1

22 What is Cloaking? free space object cloak A true cloak must have the following properties: 1. Cannot reflect or scatter waves.. 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, ransformation Electromagnetics: An Overview of the heory and Applications, IEEE Lecture 16 Ant. Prop. Mag., Vol. 5, No. 1, 4 46 (00). 43 Famous Cylinder Cloak Lecture 16 44

23 Path of Rays for Invisibility o render a region in the center of the grid invisible, the wave must be made to flow around it. Lecture Coordinate ransformation Geometry Lecture

24 Deriving the Coordinate ransformation Be observation, we wish to map our coordinates as follows. 0 R R R 1 his transform can be done with a simple straight line. y mxb m b y-intercept: b R R R1 Slope: m R 1 R R1 R R Lecture he Jacobian Matrix Due to the cylindrical geometry, the coordinate transformation will be in cylindrical coordinates. R R1 R1 R z z It follows that the Jacobian matrix is J z R R1 R z z 1 z 1z 1 1 z Lecture

25 he Material ensors he material tensors are r ' J r J det J r ' J r J det J Assuming we start with free space, we will have r R1 0 0 r r r' r' 0 0 r R 1 R r R R R1 r Lecture Cylindrical ensors rr r' r' rr r' r' zz r' r' zz 0 0 r R 1 r r r R 1 R r R1 R R1 r R 0.10 R Lecture

26 Physical Device r' r' Duke University Lecture Cartesian ensors xx 5 xy xz yx yy 5 yz zx 3 zy 3 zz Lecture

27 Carpet Cloaking Problems with Cloaking by ransformation Electromagnetics he resulting materials: Are anisotropic Require dielectric and magnetic properties Require extreme and singular values Particularly problematic at optical frequencies Lecture

28 hree Cases for Cloaking Squish an object to a point. Squish an object to a line. Squish an object to a sheet. Object becomes infinitely conducting. his is not a problem because the objects have zero size. hese can be rendered invisible, but require extreme and singular values as well as being anisotropic. 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. Lecture Carpet Cloak Concept J. Li, J. B. Pendry, Hiding under the Carpet: A New Strategy for Cloaking, Phys. Rev. Lett. 101, (008). Lecture

29 he Jacobian Matrix and Covariant Matrix We define the Jacobian matrix [J] just like we did before. A ij xi x i he Covariant matrix is then g J J Lecture he Original System ref 1 ref Lecture

30 he ransformed System Lecture he ransformation ref det g JJ detg ref ref anything 1 Lecture

31 A Preliminary ensor Description he permeability is still a tensor quantity. For a given wave, it will have two principle values. and L his gives us two refractive indices to characterize the medium. n n L L We can characterize the anisotropy using the anisotropy factor. n n L max, nl n 1 Lecture L Isotropic Medium By picking a suitable transform, we can neglect the permeability. ref det g he optimal transformation is generated by minimizing the Modified Liao functional 1 dx hw 0 0 r det g w h g dy his leads to high and low anisotropy. Lecture

32 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. quasiconformal grid ref Lecture Example Carpet Cloak Scattering from cloaked object. Scattering from just the object. Lecture

33 Other Applications Electromagnetic Cloaking of Antennas (1 of ) 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. his 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

34 Electromagnetic Cloaking of Antennas ( of ) Antenna A 1 transmits at frequency f 1. Antenna A transmits at frequency f. f 1 A 1 f A Cloak of A 1 must be transparent to f 1. Cloak of A must be transparent to f. A 1 transmitting through uncloaked A A 1 transmitting through cloaked A Lecture Reshaping the Scattering of Objects Scattering of an elliptical shaped object. Rectangular object embedded in an anisotropic medium designed by EM to scatter like the elliptical object. O. Ozgun, M. Kuzuoglu, Form Invariance of Maxwell s Equations: he Pathway to Novel Metamaterial Specifications for Electromagnetic Reshaping, IEEE Ant. And Prop. Mag., Vol. 5, No. 3, (010). Lecture

35 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 Lecture Polarization Rotator Lecture

36 Wave Collimator Lecture Flat Lenses Lecture

37 Beam Benders Lecture Concluding Remarks 37

38 Notes ransformation electromagnetics (EM) is a coordinate transformation technique where the transformation is absorbed into the permittivity and permeability functions. he method produces the permittivity and permeability as a function of position. he method does not say anything about how the materials will be realized. he resulting material functions will be anisotropic and require dielectric and magnetic metamaterials with often extreme values. his leads to structures requiring metals. EM is perhaps more commonly called transformation optics (O) because that is where it first appeared. Lecture Conformal Mapping 38

39 Observations About O Observation #1 Sharper bends lead to more extreme material properties. Observation # Oblique coordinates lead to anisotropic material properties. Least extreme Most extreme Isotropic Weakly anisotropic Strongly Anisotropic How to we always enforce this? Lecture Complex Numbers as Coordinates Imu Reu Lecture

40 Coordinate ransform with Complex Numbers u v L u v Observe the constant orthogonality of the coordinate axes. Conformal mapping Lecture