# Thick Lens Equation – Trigonometric Derivation and Matrix Formation

The following set of notes presents first a trigonometric derivation of the thick lens equation using principles such as Snell’s law and the paraxial approximation. A final formula for the thick lens equation is rather unwieldy. A matrix form is much more usable, we will find. Moreover, a matrix form allows for one to add a number of lenses together in series with ease. Parameters of the lenses can be altered as well. Soon, the matrix formation of these equations will be used in MATLAB to demonstrate the ease at which an optical system can be built using matrix formations. The matrix formation of the thick lens equation can be summarized as three matrices multiplied, for the first curved surface, the separation between the next curved surface and the final curved surface. By altering the radius of curvature, the refractive indexes at each position, distances between them using these matrices, a new lens can also be made, such as a convex thin lens by inverting the curvature of the lens and reducing the thickness on the lens. A second lens can be added in series. Once a matrix formation is made handy, there are numerous applications that then become simple.     