Exact Equations for the Back and Effective Focal Lengths of a Plano-Concave Thick Lens

In this work, we present the exact equations for the back and effective focal lengths of a plano-concave thick lens. These equations are derived through a detailed ray-tracing approach, calculating the meridional ray’s path parallel to the optical axis while taking into account lens thickness, refractive index, and surface curvature. Our analysis bridges the gap between paraxial approximations and exact ray behavior, highlighting how traditional models (e.g., thin-lens equations) can be extended to thick lenses. Additionally, we review recent advancements in negative refraction and surface phase manipulation techniques to demonstrate how our equations compare to modern numerical methods and metacoating-enhanced designs. This work provides a comprehensive and exact framework for understanding the optical properties of plano-concave thick lenses, offering new insights into their application in precision optics.