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

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

4.  Consider a series of co-central spherical shells with ε n, at the nth shell, and the radius between the nth and n+1th level is R n,n+1.

5.  We see a simple example first. We apply a uniform field E=E 0 e x, and then solve the Laplace equation in the spherical coordinate, we got solutions for the 1 st order inducing field and boundary conditions, at r = R n-1,n

6.  Then we would easily manifest A n and B n in terms of A n-1 and B n-1 as

7.  and further in matrix form where

8.  The matrix is called the transfer matrix for the n-1,n th level. If at the 1 st level there is A 1 and B 1 =0( to ensure converge) and at the infinite space there is A n = -E 0 and B n, multiply the transfer matrix again and again we will get And surly we got the solution of A 1 and B n, then whichever A k and B k you want could be solved by using transfer matrix.

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

10.  We now start some general solution for general conditions, solutions to be we apply the same B.C and trick in calculation

11.  the TMM notation will be and elements when l=1, the results would automatically turn to the same results in the previous story.

12.  Furthermore, when adding up free boundary charge, the boundary condition will turn to be we will soon get a solution no more complex than before where But the additional term, called charge term, is not that neat.

13.  Though tough, but physics For 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 got It 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.

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

15.  Consider a multi-layer withε n in the nth layer, and position of the boundary between nth and n+1th is d n,n+1 along z direction. Assume the wave propagates along z direction, perpendicular to the layer, with the expression of field, where Solve, we got As we have assumed there is not any surface current, so the boundary conditions are

16.  Then got the solution Simply we can change the expression into matrix

17.  and in it, we define If there are n layer (noticing that outside the layers are air, so the 0 th and n+1th layer are absolutely air terms), the final solution can be written as

18.  Further if we define the starting terms as, the solution could be simplified to the transfer matrix By 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.

19.  Interestingly, from the result above, one could think that, if Q 21 = 0, the reflective terms would be 0, and further if Q 11 = 1, meaning no absorption, t=1, the transmittance behavior would be perfect.  Specifically, we could input some data asε 1 =1000, ε 2 = -2000, d 1 =d 2 = 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.