Transfer Matrix Method (TMM) - EMPossible

1 9/17/2019. Advanced Computation: Computational Electromagnetics Transfer Matrix Method (TMM). 1. Outline One dimensional structures in electromagnetics Formulation of 4 4 Matrix equation for 1D structures Solution in an LHI layer Transfer matrices for multilayer structures Stability of Transfer matrices Formulation of 2 2 Matrix equation for 1D structures Slide 2. 2. 1. 9/17/2019. One Dimensional Structures in Electromagnetics Slide 3. 3. 1D Structures Sometimes it is possible to describe a physical device using just one dimension. Doing so dramatically reduces the numerical complexity of the problem and is ALWAYS GOOD. PRACTICE. x Region I.

2 Reflection Region z y Region II. Transmission Region Slide 4. 4. 2. 9/17/2019. 3D 1D Using Homogenization Many times it is possible to approximate a 3D device in one dimension. It is very good practice to at least perform the initial simulations in 1D and only moving to 3D to verify the final design. 1 2 3 4 r Physical Device Effective Medium Approximation in 1D Slide 5. 5. 3D 1D Using Circuit Wave Equivalence ni r,i r,i i Z i i . i Slide 6. 6. 3. 9/17/2019. Formulation of 4 4 Matrix Equation for 1D Structures Slide 7. 7. Starting Point Start with Maxwell's equations in the following form. Here, isotropic materials are assumed and the positive sign convention is used for waves.

3 Ez E y H z H y k0 r H x k 0 r E x y z y z Ex Ez H x H z k0 r H y k 0 r E y z x z x E y Ex H y H x k0 r H z k 0 r E z x y x y . H j 0 H Positive sign convention Slide 8. 8. 4. 9/17/2019. Calculation of the Wave Vector Components The components kx and ky are determined by the incident wave and are continuous throughout the 1D device. The kz component is different in each layer and calculated from the dispersion relation in that layer. k x k x ,inc k0 r,inc r,inc sin cos . k y k y ,inc k0 r,inc r,inc sin sin . k z ,i k02 r,i r,i k x2 k y2. i Layer #. Slide 9. 9. kx and ky Continuous Throughout Device k x ,inc k0 nair cos sin kx kx k z2,air k0 nair k x2.

4 2. k z ,inc k0 nair cos kz,air -kz,air . kinc kref x . k1 kz,1 k z2,1 k0 n1 k x2. 2. n1. kx . k2 kz,2. k z2,2 k0 n2 k x2. 2. n2 kx . k3 kz,3 k z2,3 k0 n3 k x2. 2. n3. kx . ktrn kz,air k z2,trn k0 nair k x2 k z2,air 2. kx z Slide 10. 10. 5. 9/17/2019. Waves in Homogeneous Media A wave propagating in a homogeneous layer is a plane wave. It has the following mathematical form.. E r E0 e jk r E0 e jk x x e y e jk z z H r H 0 e jk r H 0 e jk x x e y e jk z z jk y jk y Note: e+jkz sign convention was used for propagation in +z direction. When derivatives of these solutions are calculated, we see that jk y . x E r .. x E e 0. jk x x e jk y y . e jkz z jk x E0 e y e jkz z e jkx x jk x E r.

5 X jk x jk y . y E r .. y E e 0. jk x x e jk y y . e jk z z jk y E0 e y e jk z z e jk x x jk y E r .. y jk y It cannot be said that z jk z because the structure is not . homogeneous in the z direction. jk z z Slide 11. 11. Reduction of Maxwell's Equations to 1D. Given that Ez E y dE y jk x jk y k0 r H x jk y Ez k0 r H x x y y z dz Ex Ez dEx k0 r H y jk x Ez k0 r H y Maxwell's equations become z x dz E y Ex k0 r H z jk x E y jk y Ex k0 r H z x y Note: z is the only independent variable H z H y dH y left so its derivative is ordinary. k 0 r E x jk y H z k0 r E x y z dz d H x H z dH x z dz z . x k 0 r E y jk x H z k0 r E y dz H y H x k 0 r E z jk x H y jk y H x k0 r Ez x y Slide 12.

6 12. 6. 9/17/2019. Normalize the Parameters Normalize the coordinates (x, y, and z). and wave vector components (kx, ky, dE y dE y jk y Ez k0 r H x jk y Ez r H x and kz) according to dz dz . dEx dEx jk x Ez k0 r H y jk x Ez r H y z k0 z dz dz . jk x E y jk y Ex k0 r H z jk x E y jk y Ex r H z k ky k k x x k y k z z k0 k0 k0. dH y dH y jk y H z k0 r Ex jk y H z r Ex dz dz . Using the normalized parameters, dH x dH x jk x H z r E y Maxwell's equations become jk x H z k0 r E y dz dz . jk x H y jk y H x k0 r Ez jk x H y jk y H x r Ez Slide 13. 13. Solve for the Longitudinal Components Ez and Hz Solve the third and sixth equations for the longitudinal field components Hz and Ez.

7 DE y jk y Ez r H x dz . dE x jk x Ez r H y dz . jk x E y jk y Ex r H z H z . j .. k x E y k y Ex r . dH y jk y H z r Ex dz . dH x jk x H z r E y dz . jk x H y jk y H x r Ez Ez . r . j . k x H y k y H x . Slide 14. 14. 7. 9/17/2019. Eliminate the Longitudinal Components Eliminate the longitudinal field terms by substituting them back into the remaining equations. dE. jk y Ez y r H x dE y dz k y2 H x k x k y H y r r r H x dEx dz . jk x Ez r H y dz dE. r x k x2 H y k x k y H x r r H y H z . j .. k x E y k y Ex dz . r dH y jk y H z r Ex dH y dz k y2 Ex k x k y E y r r r Ex dH x dz . jk x H z r E y dz dH x 2. r k x E y k x k y Ex r r E y Ez . r.

8 J . k x H y k y H x dz . Slide 15. 15. Rearrange Maxwell's Equations Rearrange the terms and the order of the equations. dEx k x k y k 2 . dE y H x r x H y k y2 H x k x k y H y r r r H x dz r r . dz . dE dE y k y2 k x k y . r x k x2 H y k x k y H x r r H y r H x Hy dz dz r . r dH y dH x k x k y k 2 . k y2 Ex k x k y E y r r r Ex Ex r x E y dz dz r r . dH x 2. r k x E y k x k y Ex r r E y dH y k y2 k x k y dz r Ex Ey dz r . r Slide 16. 16. 8. 9/17/2019. Matrix Form of Maxwell's Equations The remaining four equations can be written in Matrix form as dEx k x k y k 2 . H x r x H y dz r r . dE y k y 2. k k . r H x x y H y . dz r r . k x k y k x2.

9 DH x k x k y . k2 . Ex r x E y 0 0 r . dz r r r r . k x k y k y2 k x k y Ex . Ex . 2. dH y ky r Ex E 0 0 r .. r E y .. dz r r y d Ey r . dz H x k x k y k x2 Hx . r 0 0 H . H y r r y .. k y2 k x k y . r 0 0 . r r . Slide 17. 17. BTW for Fully Anisotropic Materials yz yz zy k x k y yz zx k 2 yz zy . j k y k x zx jk x yx x yy . zz zz zz zz zz zz zz zz .. Ex jk xz zx zy k y2 k x k y xz zy Ex . y j k x xz k y xx xz zx xy . zz zz zz zz E y . Ey zz zz zz zz .. z H x k x k y k x2 . H . yx yz zx yy yz zy j k y yz k x zx jk x yz zy x . H. y zz zz H y . zz zz zz zz zz zz . k 2 k x k y . y xx xz zx xy xz zy jk y xz zx j k x xz k y zy . zz zz zz zz zz zz zz.

10 Zz . Note: This is for the sign convention. Slide 18. 18. 9. 9/17/2019. Solution in an LHI Layer Slide 19. 19. Matrix Differential Equation Maxwell's equations can now be written as a single Matrix differential equation. d . 0. dz . k x k y k x2 . 0 0 r . r r . Ex z k 2. k k . 0 0 y r x y . E y z r r . z . H x z k x k y k x2 . r 0 0 . H y z . r r . k y2 k x k y . r 0 0 . r r . Slide 20. 20. 10. 9/17/2019. Solution of the Differential Equation (1 of 3). The Matrix differential equation is dEx k x k y k 2 . H x r x H y dz r r . d . 0 dE y k y2 k x k y dz r H x . dz r r H y . This is actually a set of four coupled differential equations. dH x k x k y k 2.