Up to this point, the propagation algorithms have been designed to simulate propagation through vacuum and through simple optical systems that can be described by ray matrices. There are several other more complicated and useful applications of the split-step beam propagation method. These include sources with partial temporal and spatial coherence, coherent propagation through deterministic structures like fibers and integrated optical devices, and propagation through random media like atmospheric turbulence. This chapter focuses on coherent propagation through turbulence, and the method is shown to be very closely related to propagation through vacuum.

Earth's atmosphere is a medium whose refractive index is nearly unity. This allows us to make only a slight modification to our vacuum-propagation techniques from Ch. 8 to simulate propagation through the atmosphere. Unfortunately, the atmosphere's refractive index randomly evolves over space and time. This effect causes light to be randomly distorted as it propagates. As a result, optical systems that rely on light propagating through the atmosphere must overcome a great challenge. For example, astronomers have observed for centuries that atmospheric turbulence limits the resolution of their telescopes. This is why observatories are built on mountain tops; the location minimizes the turbulent path distance through which the light must propagate.

To simulate atmospheric propagation, we first develop the simulation algorithm, and then we discuss atmospheric turbulence and how to model its refractive properties. Finally, we discuss setting up an atmospheric simulation, proper sampling with due consideration to the effects of the atmosphere, and verifying that the output is consistent with analytic theory.