More complex optical systems can be analyzed by using the same methods applied to the optical systems of a single thin lens. However, the number of integrations grows as the number of free-space regions grows, and the complexity of the calculations increases as the number of lenses included grows. For these reasons, some readers may appreciate the introduction of a certain “operator” notion that is useful in analyzing complex systems. Not all readers will find this approach attractive, and for those, the methods already used can simply be extended to more complex systems.