upml free-space

Hi, how am I supposed to modify update equation when I'm using inhomogeneous media?
i.e. let say that my lattice is half free space and half dielectric. How do I need to modify update equation and UPML equation?

I summarize how to do this in my PhD dissertation in Chapter 4. Specifically, see section 4.2.2 starting on page 90. The link to this file is

https://www.edaboard.com/download.php?id=124081

-Tip