SCORPION Solar Cell: Formalism for electromagnetic waves

Below is described how the transfer matrix is applied to electromagnetic waves (for example light) of a given frequency propagating through a stack of layers at normal incidence. It can be generalized to deal with incidence at an angle, absorbing media, and media with magnetic properties. We assume that the stack layers are normal to the $z\,$ axis and that the field within one layer can be represented as the superposition of a left- and right-traveling wave with wave number $k\,$ ,

$E(z) = E_r e^{ikz} + E_l e^{-ikz}\,$ .

Because it follows from Maxwell's equation that $E\,$ and $F=dE/dz\,$ must be continuous across a boundary, it is convenient to represent the field as the vector $(E(z),F(z))\,$ , where

$F(z) = ik E_r e^{ikz} - ik E_l e^{-ikz}\,$ .

Since there are two equations relating $E\,$ and $F\,$ to $E_r\,$ and $E_l\,$ , these two representations are equivalent. In the new representation, propagation over a distance $L\,$ into the positive $z\,$ direction is described by the matrix

$M = \left( \begin{array}{cc} \cos kL & \frac{1}{k} \sin kL \\ -k \sin kL & \cos kL \end{array} \right),$

and

$\left(\begin{array}{c} E(z+L) \\ F(z+L) \end{array} \right) = M\cdot \left(\begin{array}{c} E(z) \\ F(z) \end{array} \right)$

Such a matrix can represent propagation through a layer if $k\,$ is the wave number in the medium and $L\,$ the thickness of the layer: For a system with $N\,$ layers, each layer $j\,$ has a transfer matrix $M_j\,$ , where $j\,$ increases towards higher $z\,$ values. The system transfer matrix is then

$M_s = M_N \cdot \ldots \cdot M_2 \cdot M_1.$

Typically, one would like to know the reflectance and transmittance of the layer structure. If the layer stack starts at $z=0\,$ , then for negative $z\,$ , the field is described as

$E_L(z) = E_0 e^{ik_Lz} + r E_0 e^{-ik_Lz}\,$ ,

where $E_0\,$ is the amplitude of the incoming wave, $k_L\,$ the wave number in the left medium, and $r\,$ is the amplitude (not intensity!) reflectance coefficient of the layer structure. On the other side of the layer structure, the field consists of a right-propagating transmitted field

$E_R(z) = t E_0 e^{ik_R z}\,$ ,

where $t\,$ is the amplitude transmittance and $k_R\,$ is the wave number in the rightmost medium. If $F_L = dE_L/dz\,$ and $F_R = dE_R/dz\,$ , then we can solve

$\left(\begin{array}{c} E(z_R) \\ F(z_R) \end{array} \right) = M\cdot \left(\begin{array}{c} E(0) \\ F(0) \end{array} \right)$

in terms of the matrix elements $M_{mn}\,$ of the system matrix $M_s\,$ and obtain

$t = 2 i k_L e^{-i k_R L}\left[\frac{M_{11} M_{22} - M_{12} M_{21}}{-M_{21} + k_L k_R M_{12} + i(k_R M_{11} + k_L M_{22})}\right]$

and

$r = \left[\frac{ (M_{21} + k_L k_R M_{12}) + i(k_L M_{22} - k_R M_{11})}{(-M_{21} + k_L k_R M_{12}) + i(k_L M_{22} + k_R M_{11})}\right]$ .

[edit] Example

As an illustration, consider a single layer of glass with a refractive index n and thickness d suspended in air at a wave number k (in air). In glass, the wave number is $k'=nk\,$ . The transfer matrix is

$M=\left(\begin{array}{cc}\cos k'd & \sin(k'd)/k' \\ -k' \sin k'd & \cos k'd \end{array}\right)$ .