Transfer Matrix (MATLAB)

Transfer matrix methods for plane wave transmission in multi-layer structures.

Theoretical formulation

The field in these multi-layer structures can be written in a superposition of planes waves propagating in directions $\pm z$

$$ E(z)=A_1 e^{i ((z-z_1) k_{1, z})}+B_1 e^{-i ((z-z_1) k_{1, z})},z<z_1\\ E(z)=A_m e^{i (k_{m,z} (z-z_m))}+B_m e^{-i (k_{m,z} (z-z_m))},z_{m-1}<z<z_m\\ E(z)=A_{N+1}^{`}e^{i ((z-z_1) k_{N+1,z})}+B_{N+1}^{'}e^{-i ((z-z_N) k_{N+1,z})} $$

The field coefficients can be related via boundary conditions

$$ \left( \begin{array}{c} A_{m-1} \\ B_{m-1} \\ \end{array} \right)=D_{m-1}^{-1}D_{m} \left( \begin{array}{c} A_{m}^{'} \\ B_{m}^{'} \\ \end{array} \right)= D_{m-1}^{-1}D_{m}P_{m} \left( \begin{array}{c} A_{m} \\ B_{m} \\ \end{array} \right),m=1,2,3 $$

where

$$ D_m^{\text{TE}}=\left( \begin{array}{cc} 1 & 1\\ \text{cos$\theta $}_m \sqrt{\frac{\epsilon _m}{\mu _m}}&\text{cos$\theta $}_m \left(-\sqrt{\frac{\epsilon _m}{\mu _m}}\right) \\ \end{array} \right) $$

$$ D_m^{\text{TM}}=\left( \begin{array}{c} \text{cos$\theta $}_m & \text{cos$\theta $}_m\\ n_m & -n_m\\ \end{array} \right) $$

$$ P_m=\left( \begin{array}{cc} e^{d_m \left(-\text{ik}{\text{mx}}\right)} & 0 \ 0 & e^{d_m \text{ik}{\text{m x}}} \end{array} \right) $$

Therefore, the field relation between field components in left and right space are

$$ \begin{aligned} & \left(\begin{array}{l} A_1 \\ B_1 \end{array}\right)=D_1^{-1} D_2\left(\begin{array}{l} A_2^{\prime} \\ B_2^{\prime} \end{array}\right)=D_1^{-1} D_2 P_2\left(\begin{array}{l} A_2 \\ B_2 \end{array}\right) \\ & =D_1^{-1} D_2 P_2 D_2^{-1} D_3 P_3 \ldots D_N P_N D_N^{-1} D_{N+1}\left(\begin{array}{l} A_{N+1}^{\prime} \\ B_{N+1}^{\prime} \end{array}\right) \\ & =M_{1 \text { toN+1 }}\left(\begin{array}{l} A_{N+1}^{\prime} \\ B_{N+1}^{\prime} \end{array}\right) \end{aligned} $$

The general boundary conditions should be

$$ A_1=1, B_{N+1}^{}=0, B_1=1, A_{N+1}^{}=0 $$

References: TRANSFER MATRIX APPROACH TO PROPAGATION OF ANGULAR PLANE WAVE SPECTRA THROUGH METAMATERIAL MULTILAYER STRUCTURES, Han Li, UNIVERSITY OF DAYTON, Thesis

CoeAB_layer_TMM: This function calculates the field coefficients in different layers.

field_layer_from_ABCoe: This function calculates the exact field distributions with given AB coefficients.

Examples

Transmission and Reflections Coefficients of Brag Mirror

BragMirror_1D.m

Can be compared with the RCWA methods

Field in photoresist

BenchMark_REF1985.m_

This program simulates the field in photoresist and can be compared with reference [2]:

[2]Mack, C. A. Analytical expression for the standing wave intensity in photoresist. Appl. Opt., AO 25, 1958–1961 (1986).