The analogy is very bad. Yes, there are formulae and they are simpler. It is sufficient to make a google search. There is a lot of material on the topic.
the anology between them is not perfect, i've a document for internet but it is in frensh and formulas are the same in two languages ... it can help you :
There are formulas that describe the properties of photonic crystals, however, it is important to realize that the formulas cannot be solved analytically, but need to be solved numerically, at least for the 2D and 3D case.
You can find a code that solves for photonic crystals slabs (most of the relevant figures of merit, e.g. loss, group velocity, frequency bands etc..) here: