Theory of the transfer matrix method

Schematic of the Transfer Matrix Method showing forward and backward propagating electromagnetic fields in a multilayer optical stack — Schematic representation of the Transfer Matrix Method (TMM) for a multilayer optical structure. Each layer is described by a complex refractive index \(n + jk\) and thickness \(l\), while forward and backward propagating electromagnetic waves (\(E^+\) and \(E^-\)) are coupled at every interface. The total optical field inside the device is obtained by propagating the fields through the full stack.

1. Introduction

On this page we derive the transfer matrix method (TMM) for multilayer dielectric structures, as shown in figure ??. Such layered structures resemble the optical stacks found in OLEDs, thin-film solar cells and optical coatings, where the individual layer thicknesses are comparable to the wavelength of light, typically ranging from hundreds of nanometres to several microns. Under these conditions, optical interference, reflection and absorption strongly influence device performance and must therefore be treated using Maxwell’s equations and wave optics rather than simple ray optics. For a broader introduction to the transfer matrix method and its applications in optical modelling, see What is the transfer matrix method?. For a practical step-by-step guide showing how to use the optical transfer matrix model in OghmaNano, see Optical transfer matrix model.

2. Derivation

Considering the interface between layers 1 and 2 in figure ??, the electric field on the left-hand side of the interface is given by

\begin{equation} E_{1}=E^{+}_{1} e^{-j k_1 z}+E^{-}_{1} e^{j k_1 z} \label{eq:efield1} \end{equation}

and on the right-hand side of the interface the electric field is given by

\begin{equation} E_{2}=E^{+}_{2} e^{-j k_2 z}+E^{-}_{2} e^{j k_2 z} \label{eq:efield2} \end{equation}

Maxwell’s equations give the relationship between the electric and magnetic fields for a plane wave,

\begin{equation} \nabla \times \mathbf{E}=-j\omega \mu \mathbf{H} \end{equation}

which in one dimension simplifies to

\begin{equation} \frac{\partial E} {\partial z}=-j\omega \mu H \label{eq:maxwell} \end{equation}

Applying equation \eqref{eq:maxwell} to equations \eqref{eq:efield1} and \eqref{eq:efield2}, we obtain the magnetic field on the left-hand side of the interface

\begin{equation} -j \mu \omega H^{y}_{1}=-j k_1 E^{+}_{1} e^{-j k_1 z}+j k_1 E^{-}_{1} e^{j k_1 z} \end{equation}

and on the right-hand side of the interface

\begin{equation} -j \mu \omega H^{y}_{2}=-j k_2 E^{+}_{2} e^{-j k_2 z}+j k_2 E^{-}_{2} e^{j k_2 z} \end{equation}

Tidying up gives,

\begin{equation} H^{y}_{1}=\frac{k_1}{\omega \mu}E^{+}_{1} e^{-j k_1 z}-\frac{k_1}{\omega \mu} E^{-}_{1} e^{j k_1 z} \end{equation} \begin{equation} H^{y}_{2}=\frac{k_2}{\omega \mu}E^{+}_{2} e^{-j k_2 z}-\frac{k_2}{\omega \mu} E^{-}_{2} e^{j k_2 z} \end{equation}

3. Boundary conditions

We now apply the electric and magnetic boundary conditions

\begin{equation} \mathbf{n} \times (\mathbf{E_2}-\mathbf{E_1})=0 \end{equation} \begin{equation} \mathbf{n} \times (\mathbf{H_2}-\mathbf{H_1})=0 \end{equation}

Evaluating the fields at the interface \(z=0\) gives

\begin{equation} (E_{2}^{+}+E_{2}^{-})-(E_{1}^{+}+E_{1}^{-})=0 \label{eq:electric_boundary} \end{equation}

and

\begin{equation} \frac{k_2}{\omega \mu}(E_{2}^{+}-E_{2}^{-}) - \frac{k_1}{\omega \mu}(E_{1}^{+}-E_{1}^{-})=0 \end{equation}

The wavevector is given by

\begin{equation} k=\frac{2\pi n}{\lambda} \end{equation}

We can therefore write the magnetic boundary condition as

\begin{equation} n_2 (E_{2}^{+}-E_{2}^{-}) - n_1 (E_{1}^{+}-E_{1}^{-})=0 \label{eq:mag_boundary} \end{equation}

4. Forward propagating wave

Rearranging equation \eqref{eq:mag_boundary} gives

\begin{equation} E_{1}^{-} = E_{1}^{+} - \frac{n_2}{n_1}(E_{2}^{+}-E_{2}^{-}) \end{equation}

Substituting into equation \eqref{eq:electric_boundary} gives

\begin{equation} E_{2}^{+}+E_{2}^{-} = E_{1}^{+}+E_{1}^{+} - \frac{n_2}{n_1}(E_{2}^{+}-E_{2}^{-}) \end{equation} \begin{equation} 2E_{1}^{+} = E_{2}^{+}+E_{2}^{-} + \frac{n_2}{n_1}(E_{2}^{+}-E_{2}^{-}) \end{equation} \begin{equation} 2E_{1}^{+}\frac{n_1}{n_1+n_2} = E_{2}^{+} + E_{2}^{-}\frac{n_1-n_2}{n_1+n_2} \end{equation}

5. Backwards propagating wave

Rearrange equation \eqref{eq:mag_boundary} to give,

\begin{equation} E_{1}^{+}=E_{1}^{-} +\frac{n_2}{n_1}(E_{2}^{+}-E_{2}^{-}) \end{equation}

Inserting in equation \eqref{eq:electric_boundary}, gives

\begin{equation} E_{2}^{+}+E_{2}^{-}=E_{1}^{-} +\frac{n_2}{n_1}(E_{2}^{+}-E_{2}^{-})+E_{1}^{-} \end{equation} \begin{equation} 2E_{1}^{-}=E_{2}^{+}+E_{2}^{-}- \frac{n_2}{n_1}(E_{2}^{+}-E_{2}^{-}) \end{equation} \begin{equation} 2E_{1}^{-}\frac{n_1}{n_1+n_2}=E_{2}^{+}\frac{n_1-n_2}{n_1+n_2}+E_{2}^{-} \end{equation}

These equations become:

\begin{equation} E_{1}^{-}t_{12}=E_{2}^{+}r_{12}+E_{2}^{-} \end{equation}

and

\begin{equation} E_{1}^{+}t_{12}=E_{2}^{+}+E_{2}^{-}r_{12} \end{equation}

Accounting for propagation we can write.

\begin{equation} E_{1}^{+}t_{12}=E_{2}^{+}e^{\zeta_2 d_1}+E_{2}^{-}r_{12}e^{-\zeta_2 d_1} \end{equation}

and

\begin{equation} E_{1}^{-}t_{12}=E_{2}^{+}r_{12}e^{\zeta_2 d_1}+E_{2}^{-}e^{-\zeta_2 d_1} \end{equation}

where

\begin{equation} \zeta=\frac{2\pi}{\lambda} \bar{n} \end{equation}

6. Transfer matrix implementation

The transfer matrix method can now be implemented by writing the optical fields in each layer as a vector containing the forward and backward propagating waves. The multilayer optical stack is then solved by repeatedly multiplying small matrices describing propagation through each layer and reflection/transmission at each interface. This approach is computationally efficient and only requires simple matrix multiplication operations, making it straightforward to implement numerically.

The optical field in layer \(i\) is written as

\begin{equation} \begin{pmatrix} E_i^{+} \\ E_i^{-} \end{pmatrix} \end{equation}

where \(E_i^{+}\) and \(E_i^{-}\) are the forward and backward propagating waves respectively.

At each interface the Fresnel reflection and transmission coefficients are given by

\begin{equation} r_{ij}=\frac{n_i-n_j}{n_i+n_j}, \qquad t_{ij}=\frac{2n_i}{n_i+n_j} \end{equation}

The interface between layers \(i\) and \(j\) can therefore be written as

\begin{equation} \begin{pmatrix} E_i^{+} \\ E_i^{-} \end{pmatrix} = \frac{1}{t_{ij}} \begin{pmatrix} 1 & r_{ij} \\ r_{ij} & 1 \end{pmatrix} \begin{pmatrix} E_j^{+} \\ E_j^{-} \end{pmatrix} \end{equation}

Propagation through a layer of thickness \(d_i\) is described by

\begin{equation} \begin{pmatrix} E_i^{+}(z+d_i) \\ E_i^{-}(z+d_i) \end{pmatrix} = \begin{pmatrix} e^{\zeta_i d_i} & 0 \\ 0 & e^{-\zeta_i d_i} \end{pmatrix} \begin{pmatrix} E_i^{+}(z) \\ E_i^{-}(z) \end{pmatrix} \end{equation}

where

\begin{equation} \zeta_i=\frac{2\pi \bar{n}_i}{\lambda} \end{equation}

The transfer matrix for a single layer is obtained by multiplying the interface and propagation matrices together

\begin{equation} M_i= \frac{1}{t_{ij}} \begin{pmatrix} 1 & r_{ij} \\ r_{ij} & 1 \end{pmatrix} \begin{pmatrix} e^{\zeta_i d_i} & 0 \\ 0 & e^{-\zeta_i d_i} \end{pmatrix} \end{equation}

The optical response of the full multilayer device is then obtained by multiplying the matrices together

\begin{equation} M=M_1M_2M_3 \cdots M_n \end{equation}

such that

\begin{equation} \begin{pmatrix} E_0^{+} \\ E_0^{-} \end{pmatrix} = M \begin{pmatrix} E_n^{+} \\ E_n^{-} \end{pmatrix} \end{equation}

For a device with no incoming wave from the rear contact, \(E_n^{-}=0\). The reflected and transmitted fields can then be calculated directly from the resulting matrix elements.

7. Solving in one big matrix

Rather than solving the transfer matrix equations sequentially, one can instead solve the entire multilayer optical system in one large matrix containing all forward and backward propagating waves simultaneously.

For a device with non reflecting back contacts:

\begin{equation} \begin{pmatrix} e^{\zeta d} & 0 & 0 & 0 & r_{01}e^{-\zeta d} & 0 & 0 & 0 \\ -t_{12} & e^{\zeta d} & 0 & 0 & 0 & r_{12}e^{-\zeta d} & 0 & 0 \\ 0 & -t_{23} & e^{\zeta d} & 0 & 0 & 0 & r_{23}e^{-\zeta d} & 0 \\ 0 & 0 & -t_{34} & e^{\zeta d} & 0 & 0 & 0 & r_{34}e^{-\zeta d} \\ 0 & r_{12}e^{\zeta d_1} & 0 & 0 & -t_{12} & e^{-\zeta d} & 0 & 0 \\ 0 & 0 & r_{23}e^{\zeta d} & 0 & 0 & -t_{23} & e^{-\zeta d} & 0 \\ 0 & 0 & 0 & r_{34}e^{\zeta d} & 0 & 0 & -t_{34} & e^{-\zeta d} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -t_{45} \end{pmatrix} \begin{pmatrix} E_{1}^{+} \\ E_{2}^{+} \\ E_{3}^{+} \\ E_{4}^{+} \\ E_{1}^{-} \\ E_{2}^{-} \\ E_{3}^{-} \\ E_{4}^{-} \end{pmatrix} = \begin{pmatrix} t_{01}E_{external} \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{pmatrix} \end{equation}

8. Complex refractive index and optical absorption

Optical absorption in the transfer matrix method is included through the complex refractive index \(n+j\kappa\), where \(n\) is the refractive index and \(\kappa\) is the extinction coefficient. The electric field propagating through an absorbing medium is given by

\begin{equation} E(z,t)=Re(E_0 e^{j(-kz+\omega t)})= Re(E_0 e^{j(\frac{-2 \pi (n+j\kappa)}{\lambda}z + \omega t)})=e^{\frac{2\pi\kappa z}{\lambda}}Re(E_0 e^{\frac{j(-2 \pi (n+j\kappa)}{\lambda}z +\omega t}) \end{equation}

Because the optical intensity is proportional to the square of the electric field, the absorption coefficient becomes

\begin{equation} e^{-\alpha x}=e^{\frac{2\pi\kappa z}{\lambda}} \end{equation} \begin{equation} \alpha=-\frac{4\pi\kappa}{\lambda_0} \end{equation}

👉 Next step: Continue to the Transfer Matrix simulation tool to learn how TMM is implemented inside OghmaNano.