An approximated closed-form solution for inhomogeneous planar layers

An approximated closed-form analytic solution is introduced for arbitrary inhomogeneous planar layers (IPLs). First, the differential equations of IPLs are written as a suitable matrix differential equation. Then, the matrix differential equation is solved to obtain the chain parameter matrix of IPLs. Afterwards, the electric and magnetic fields at any point and also the reflection and transmission coefficients are obtained using the chain parameter matrix. The validation of the introduced solution is studied finally.