Invisibility through the Ewald-Oseen Extinction Theorem

My understandings of this method are somewhat flimsy - so my explanation here will benefit my knowledge as well as yours. Before I continue, it is crucial that I say that anything that I mention from now on represents work done or built on that by Matthew Berg and Chris Sorenson. As this webpage is somewhat informal, I may occasionally fail to cite properly - however it shouldn't be forgotten that I could not have attempted this project were it not for both of their help. With that being said...

Recall the extinction coefficient (See Appendix). If we could somehow make the extinction coefficient zero, then light would effectively not interact with whatever substance we were working with, making it invisible.

So, theoretically, if we wanted to make a material that was invisible, i.e. the extinction coefficient zero, we would have to find a way to make both the scattered and absorbed components be equal to zero. How is this possible? Well, recall that the electromagnetic optical theorem states (Berg):

Simply by visual inspection, it is possible to tell that there are several circumstances in which the extinction cross section will be zero. The one that is most relevant to our research is when

This occurs only when the two elements of the scalar product are orthogonal to one another.