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 Mi.
i is simply a dummy variable and u and l subscripts refer to either upper or lower interfaces.
Obtain expressions for ail and bil
In matrix form:
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 Mi.
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.