Greenleaf and his collaborators used sophisticated mathematics to understand what must be happening inside the cloaked region. Everything seemed fine when they applied the Helmholtz equation, an equation widely used to solve problems involving the propagation of light. But when they used Maxwell's equations, which take the polarization of electromagnetic waves into account, difficulties came to light.
Maxwell's equations said that a simple copper disk like the one Smith used could be cloaked without a problem, but anything that emitted electromagnetic waves--a cell phone, a digital watch, or even a simple electric device like a flashlight--caused the behavior of the cloaking device to go seriously awry. The mathematics predicts that the size of the electromagnetic fields go to infinity at the surface of the cloaked region, possibly wrecking the invisibility.
Their analysis also revealed another surprise: a person trying to look out of the cloak would effectively be faced with a mirror in every direction. If you can imagine Harry Potter's own invisibility cloak working this way, and Harry turning on his flashlight to see, its light would shine right back at him, no matter where he pointed it.
Greenleaf's team determined that a more complicated phenomenon arises when using Maxwell's equations, leading to a "blow up" (an unexpected infinite behavior) of the electromagnetic fields. They determined that by inserting conductive linings, whose properties depend on the specific geometry of the cloak, this problem can be resolved. Alternatively, covering both the inside and outside surfaces of the cloaked region with carefully matched materials can also be used to bypass this problem.