Transfer Matrix Reflectivity Calculation

This section describes the derivation of the transfer matrix used in the calculation for the reflectivity spectrum. First we consider the transmission and reflection off a single layer in a multiple layer system. The Electrical and Magnetic components of the wave in the upper interface of the material must be related to the Electrical and Magnetic components in the lower interface by some transformation M_i.

i is simply a dummy variable and u and l subscripts refer to either upper or lower interfaces.

where

Obtain expressions for a_il and b_il

In matrix form:

or

where

The effect of absorption can by adding a term at the beginning of the analysis but it amounts to simply by adding the term in the argument of the cos and sin functions in each element of matrix M_i.

Next we shall use transfer matrices to calculate the reflectivity of a multilayer stack of semiconductor or dielectric materials such as might be found in a distributed Bragg reflector or the entire arrangement of layers in a VCSEL structure.