照人学 Fudan University Transfer Matrix Method In Solving EM Problem WNBOIN PO oduced by Yaoxuan Li, Weijia Wang, Shaojie Ma Presented by Y.X. Li
QUTLINE Introducing Transfer Matrix in Solving Laplace Equation 2/ General Properties for TMM in Multi-layer Shell 3 General use in EM Wave Propagating in Multi-layer
OUTLINE 1 Introducing Transfer Matrix in Solving Laplace Equation 2 General Properties for TMM in Multi-layer Shell 3 General use in EM Wave Propagating in Multi-layer
Introducing Transfer Matrix in Solving Laplace Equation 2/General Properties for TMM in Multi-layer Shell General use in EM Wave Propagating in Multi-layer
1 Introducing Transfer Matrix in Solving Laplace Equation 2 General Properties for TMM in Multi-layer Shell 3 General use in EM Wave Propagating in Multi-layer
INTRODUCING TRANSFER MATRIX IN SOLVING LAPLACE EQUATION x Consider a series of co-central spherical shells with En, at the nth shell, and the radius between the nth and n+ 1th level is R n,n+1
INTRODUCING TRANSFER MATRIX IN SOLVING LAPLACE EQUATION 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 x We see a simple example first We apply a uniform field E=Eoex, and then solve the Laplace equation in the spherical coordinate, we got solutions for the 1st order inducing field B (A, r+- n)cos 6 and boundary conditions n-1 ar O at r= R n-1
INTRODUCING TRANSFER MATRIX IN SOLVING LAPLACE EQUATION 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 1 st order inducing field and boundary conditions , at r = Rn-1,n 2 ( ) cos n n n B A r r = + n n −1 = 1 1 n n n n r r − − =
INTRODUCING TRANSFER MATRIX IN SOLVING LAPLACE EQUATION x Then we would easily manifest a and b in terms of A-, and n-1 as 2+ 2-2 B A,+ 3 R 1+2 B A,R,+ n-1.n
INTRODUCING TRANSFER MATRIX IN SOLVING LAPLACE EQUATION Then we would easily manifest An and Bn in terms of An-1 and Bn-1 as 1 1 1 1 1, 2 2 2 3 3 n n n n n n n n n B A A R − − − − − + − = + 1 1 1 1, 1 1 1 2 3 3 n n n n B A R B n n n n n − − − − − − + = +
INTRODUCING TRANSFER MATRIX IN SOLVING LAPLACE EQUATION x and further in matrix form Q1Q12 B八(Q1Q2八Bn where 2+ 2-2 Q Q12 3 R n-1,n 1+2 R in 22
INTRODUCING TRANSFER MATRIX IN SOLVING LAPLACE EQUATION and further in matrix form where 11 12 1 21 22 1 n n n n A A Q Q B B Q Q − − = 1 22 1 2 3 n n Q − + = 1 21 1, 1 3 n n Q Rn n − − − = 1 12 1, 2 2 1 3 n n n n Q R − − − = 1 11 2 3 n n Q − + =
INTRODUCING TRANSFER MATRIX IN SOLVING LAPLACE EQUATION x The matrix Qn-I is called the transfer matrix for the O n-1, n th level. If at the 1st level there is a, and b,=o( to ensure converge)and at the infinite space there is A =-Eo and B multiply the transfer matrix again and again we will get E A B On_.02. 0 And surly we got the solution of a, and B, then whichever A and Bk you want could be solved by using transfer matrix
INTRODUCING TRANSFER MATRIX IN SOLVING LAPLACE EQUATION The matrix is called the transfer matrix for the n-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 get And surly we got the solution of A1 and Bn , then whichever Ak and Bk you want could be solved by using transfer matrix. 11 12 1, 21 22 n n Q Q Q Q Q − = 0 1 , 1 1, 2 2,1 ... 0 n n n n n E A Q Q Q B − − − − =
Introducing Transfer Matrix in Solving Laplace Equation 2/ General Properties for TMM in Multi-layer Shell General use in EM Wave Propagating in Multi-layer
1 Introducing Transfer Matrix in Solving Laplace Equation 2 General Properties for TMM in Multi-layer Shell 3 General use in EM Wave Propagating in Multi-layer
GENERAL PROPERTIES FOR TMM IN MULT-LAYER SHELL x We now start some general solution for general conditions, solutions to be ∑( B )P(cos0) we apply the same B. C and trick in calculation R n.n+1 R +1 n.n+1 n.n+1 R 1,n +1 +1 En(+1) B B + 1.n n+11n,n+1 En(+1)n+2 R ,n+1 n,n+1
GENERAL PROPERTIES FOR TMM IN MULTI-LAYER SHELL We now start some general solution for general conditions, solutions to be we apply the same B.C and trick in calculation 1 ( ) (cos ) l l l n n n l l l B A r P r + = + , 1 , 1 1 1 , 1 , 1 1 1 1 ! , 1 1 , 1 1 2 2 , 1 , 1 1 1 1 1 ( 1) ( 1) l l n n n n l l l l n n n n n n l l l l n n n n n n n n n n l l n n n n R R R R A A B B lR l lR l R R + + + + + + + − − + + + + + + + + + = − + − +
