You can see the following links:

1. https://www.osapublishing.org/oe/fulltext.cfm?uri=oe-26-15-19596&id=395483

2. https://anr.fr/Project-ANR-10-JCJC-0304

if you mean the noise from using the photonic crystal as a guide.

there are many things to think about: the thicknesses of the layers can vary upon fabrication, backscattering due to lattice imperfections, attenuation from layers absorption coefficients, etc.

Is there a formula for this? For example, is phase change caused by noise in the structure?

It is hard to estimate these factors

I think you should look for experimental data

Your question is somewhat vague -- what do you mean by "state", and by "noise"?

You (e.g.) might be asking how robust the designed-for bandstructure is against imperfections in fabrication, for example. If so, the answer will depend on how accurate the design was, and (perhaps most importantly) on the size, scale, and type of fabrication errors during manufacture.

Or you might be asking about how robust their designed for behavior or performance is to noisy input or environments. Again, that will depend on the design, and also on the nature of the inputs.

But whatever question you actually intended, the answer is going to depend a great deal on what sort of design you use, what sort of materials, what sort of fabrication, and so on. You should expect that you will get very different answers, and all depending on what sort "photonic crystal" it is you mean.