Lecture 16 -- Transformation Electromagnetics
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 1 4/30/2015 Introduction Design Process Using Spatial Transforms Step 1 of 4: Define Spatial Transform Lecture 16 4 2 4/30/2015 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 3 4/30/2015 Design Process Using Spatial Transforms Step 4 of 4: Generate Overall Lattice Lecture 16 7 Coordinate Transformation 4 4/30/2015 Coordinate Transformation We can map one coordinate space into another The image part with relationship ID rId4 was not found in the file. r xxˆ yyˆ 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 5 4/30/2015 Example #1: Cylindrical to Cartesian The Cartesian and cylindrical coordinates are related through x cos y sin zz 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 12 6 4/30/2015 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 7 4/30/2015 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 8 4/30/2015 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 9 4/30/2015 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 10 4/30/2015 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 11 4/30/2015 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 12 4/30/2015 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 13 4/30/2015 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 14 4/30/2015 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 15 4/30/2015 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 32 16 4/30/2015 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 17 4/30/2015 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 18 4/30/2015 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 38 19 4/30/2015 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 20 4/30/2015 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). 42 21 4/30/2015 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 22 4/30/2015 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 46 23 4/30/2015 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 48 24 4/30/2015 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 25 4/30/2015 Cartesian Tensors Accuracy of these results is not guaranteed! Lecture 16 51 Carpet Cloaking 26 4/30/2015 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. 54 27 4/30/2015 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 28 4/30/2015 The Original System ref ref 1 Lecture 16 57 The Transformed System Lecture 16 58 29 4/30/2015 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 30 4/30/2015 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 62 31 4/30/2015 Example Carpet Cloak Scattering from cloaked object. Scattering from just the object. Lecture 16 63 Other Applications 32 4/30/2015 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 66 33 4/30/2015 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 68 34 4/30/2015 Polarization Rotator Lecture 16 69 Wave Collimator Lecture 16 70 35 4/30/2015 Flat Lenses Lecture 16 71 Beam Benders Lecture 16 72 36 4/30/2015 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 74 37
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