Transfer-matrix Method (optics) - Formalism For Electromagnetic Waves

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 axis and that the field within one layer can be represented as the superposition of a left- and right-traveling wave with wave number ,

.

Because it follows from Maxwell's equation that and must be continuous across a boundary, it is convenient to represent the field as the vector, where

.

Since there are two equations relating and to and, these two representations are equivalent. In the new representation, propagation over a distance into the positive direction is described by the matrix

and

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

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

,

where is the amplitude of the incoming wave, the wave number in the left medium, and 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

,

where is the amplitude transmittance and is the wave number in the rightmost medium. If and, then we can solve

in terms of the matrix elements of the system matrix and obtain

and

.

The transmittance and reflectance (i.e., the fractions of the incident intensity transmitted and reflected by the layer) are often of more practical use and are given by and, respectively (at normal incidence).