Tmatrix - simple, efficient implementation of Transfer Method for multilayer stacks

1. What is the Transfer Method?

It is possible to simulate the propagation of light through multilayered media using the transfer matrix method. Simply, this involves assigning each interface/layer a matrix and then multiplying these terms together.

Let us assume we have a plane wave propagating in the forward direction ($+z$):

$$ \begin{equation*} E^{+}(z) = E^{+}(0)e^{ik_{z}z} \end{equation*} $$

where in this equation $E^{+}(z)$ means the electric field of the wave is propagating to the right and $k_{z}$ is the $z$ componenent of the wavevector of the plane wave, which is equal to

$$ \begin{equation*} k_{z} = k_{0}ncos(\theta) = \frac{2\pi}{\lambda_{0}}ncos(\theta) \end{equation*} $$

where $n$ is the refractive index of the medium, $\theta$ is the angle of incidence and $\theta_{0}$ is the wavelength of the light in a vacuum.

Propagation Matrix

We can use matrices to calculate the situation in which waves are propagating both forward (defined as to the right, $+z$) and backwards (defined as to the left, $-z$) through a layer of thickness $d$:

$$ \begin{align*} E^{+}(d) &= E^{+}(0)e^{ik_{z}d} \\ E^{-}(d) &= E^{-}(0)e^{-ik_{z}d} \end{align*} $$

Alternatively we can write this in matrix form as

$$\left[ {\begin{array}{cc} E^{+}(d) \\\ E^{-}(d) \end{array} } \right] = \left[ {\begin{array}{cc} e^{ik_{z}d} & 0 \\\ 0 & e^{-ik_{z}d} \end{array} } \right]\left[ {\begin{array}{cc} E^{+}(0) \\\ E^{-}(0) \end{array} } \right]$$

Since we want subsequent matrices to be inserted on the right we will invert the form of this matrix equation to

$$\left[ {\begin{array}{cc} E^{+}(0) \\\ E^{-}(0) \end{array} } \right] = \left[ {\begin{array}{cc} e^{-ik_{z}d} & 0 \\\ 0 & e^{ik_{z}d} \end{array} } \right]\left[ {\begin{array}{cc} E^{+}(d) \\\ E^{-}(d) \end{array} } \right]$$

Let us now define the transfer (propagation) matrix as

$$T^{(prop)} = \left[ {\begin{array}{cc} e^{-ik_{z}d} & 0 \\\ 0 & e^{ik_{z}d} \end{array} } \right] = \left[ {\begin{array}{cc} e^{-ik_{0}dncos(\theta)} & 0 \\\ 0 & e^{ik_{0}dncos(\theta)} \end{array} } \right]$$

Interface Matrix

For the TE (Transverse electric; s-polarisation) and TM (Transverse magnetic; p-polarisation) cases we can calculate the reflection and transmission coeficients using the Fresnel equations

$$r^{(TE)}_{12} = \frac{n_{1}cos(\theta_{1}) - n_{2}cos(\theta_{2})}{n_{1}cos(\theta_{1}) + n_{2}cos(\theta_{2})}$$ $$t^{(TE)}_{12} = \frac{2n_{1}cos(\theta_{1})}{n_{1}cos(\theta_{1}) + n_{2}cos(\theta_{2})}$$ $$r^{(TM)}_{12} = \frac{n_{2}cos(\theta_{1}) - n_{1}cos(\theta_{2})}{n_{2}cos(\theta_{1}) + n_{1}cos(\theta_{2})}$$ $$t^{(TM)}_{12} = \frac{2n_{1}cos(\theta_{1})}{n_{2}cos(\theta_{1}) + n_{1}cos(\theta_{2})}$$

$r_{12}$ and $t_{12}$ are the complex reflection and transmission coefficients from medium 1 to 2 (for either TE or TM polarisation).

The angles $\theta_{1}$ and $\theta_{2}$ are the angles of propagation in medium 1 and 2 respectively, which are related by Snell's Law

$$n_{1}sin(\theta_{1}) = n_{2}sin(\theta_{2})$$

Using these equations we can now calculate the transfer matrix corresponding to the interface between two media. Waves coming from the left and right side will result in output waves given by the scattering matrix

$$\left[ {\begin{array}{cc} E^{-} \\\ E^{+} \end{array} } \right] = \left[ {\begin{array}{cc} r_{12} & t_{21} \\\ t_{12} & r_{21} \end{array} } \right]$$