Casimir effect force formula and plate mode frequencies

[raracon]

TL;DR

Quick fact check

"In the example of Fig. 1, Maxwell’s equations allow field modes of arbitrarily large frequency both between the plates and outside them, and therefore the zero-point field energy is infinite when the plates are separated by a finite distance d as well as when they are infinitely far apart. However, the difference in zero-point energy for the two cases is finite, and its dependence on the plate separation d implies a force F = −πhc/480d^4 per unit area."

Fig 1:

Read this in a book about the Casimir Effect. Now my question is if it is correct to say that between the two perfectly conducting plates there can be modes with any frequency? I've read in other books that the frequency is limited by the separation between the plates.

And my other question is how does one explain F = −πhc/480d^4 ?

 

[anuttarasammyak]

A Wiki article https://en.wikipedia.org/wiki/Casimir_effect shows formula, the excitation is quantized by plane boundary condition so lower vacuum energy than outside take place between the plates. The formula of F is also derived there.