Transfer Matrix Method for Solving EM Problems in Multi-Layer Structures

1. Transfer Matrix Method In Solving EM Problem Produced by Yaoxuan Li , Weijia Wang , Shaojie Ma Presented by Y.X.Li

2. 4 Theoretical Analyzing Introducing Transfer Matrix in Solving Laplace Equation 1 2 General Properties for TMM in Multi-layer Shell 3 General use in EM Wave Propagating in Multi-layer outline

3. 4 Theoretical Analyzing Introducing Transfer Matrix in Solving Laplace Equation 1 2 General Properties for TMM in Multi-layer Shell 3 General use in EM Wave Propagating in Multi-layer

4. Consider a series of co-central spherical shells with εn, at the nth shell, and the radius between the nth and n+1th level is Rn,n+1. Introducing Transfer Matrix in Solving Laplace Equation

5. We see a simple example first. We apply a uniform field E=E0ex , and then solve the Laplace equation in the spherical coordinate, we got solutions for the 1st order inducing fieldand boundary conditions , at r = Rn-1,n Introducing Transfer Matrix in Solving Laplace Equation

6. Then we would easily manifest An and Bn in terms of An-1 and Bn-1 as Introducing Transfer Matrix in Solving Laplace Equation

7. and further in matrix form where Introducing Transfer Matrix in Solving Laplace Equation

8. The matrix is called the transfer matrix for then-1,n th level. If at the 1st level there is A1 and B1=0( to ensure converge) and at the infinite space there is An= -E0 and Bn, multiply the transfer matrix again and again we will getAnd surly we got the solution of A1 and Bn, then whichever Ak and Bk you want could be solved by using transfer matrix. Introducing Transfer Matrix in Solving Laplace Equation

9. 4 Theoretical Analyzing Introducing Transfer Matrix in Solving Laplace Equation 1 2 General Properties for TMM in Multi-layer Shell 3 General use in EM Wave Propagating in Multi-layer

10. We now start some general solution for general conditions, solutions to be we apply the same B.C and trick in calculation General Properties for TMM in Multi-layer Shell

11. the TMM notation will be and elementswhen l=1, the results would automatically turn to the same results in the previous story. General Properties for TMM in Multi-layer Shell

12. Furthermore, when adding up free boundary charge, the boundary condition will turn to bewe will soon get a solution no more complex than before whereBut the additional term, called charge term, is not that neat. General Properties for TMM in Multi-layer Shell

13. Though tough, but physicsFor a metal layer, the potential is constant, therefore only , otherwise is 0. If in the boundary for n,n+1th layer there is a l order charge ,we would directly gotIt means, the surface charge density is just like other external conditions such as E field, would only induce the same order term, as a uniform E inducing only a term. General Properties for TMM in Multi-layer Shell

14. 4 Theoretical Analyzing Introducing Transfer Matrix in Solving Laplace Equation 1 2 General Properties for TMM in Multi-layer Shell 3 General use in EM Wave Propagating in Multi-layer

15. Consider a multi-layer withεn in the nth layer, and position of the boundary between nth and n+1th is dn,n+1 along z direction. Assume the wave propagates along z direction, perpendicular to the layer, with the expression of field, where Solve , we gotAs we have assumed there is not any surface current, so the boundary conditions are General use in EM Wave Propagating in Multi-layer

16. Then got the solutionSimply we can change the expression into matrix General use in EM Wave Propagating in Multi-layer

17. and in it, we defineIf there are n layer (noticing that outside the layers are air, so the 0th and n+1th layer are absolutely air terms), the final solution can be written as General use in EM Wave Propagating in Multi-layer

18. Further if we define the starting terms as ,the solution could be simplified to the transfer matrixBy using this method we could quickly get the answer of t and r And from this we could use transfer method to get the propagating properties in any layers. General use in EM Wave Propagating in Multi-layer

19. Interestingly, from the result above, one could think that, if Q21= 0, the reflective terms would be 0, and further if Q11= 1, meaning no absorption, t=1 , the transmittance behavior would be perfect. Specifically, we could input some data asε1=1000, ε2= -2000, d1=d2= 2mm, then we get ω= 2π*0.850 GHz, there is a perfect transmission. From a COMSOL simulation we can see the S21 is near to 1. Comsol simulation