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Using simulation software (CST Studio), I calculated the dispersion of the eigenmodes of a photonic crystal, that is frequency vs wavenumber for the desired propagation direction. I wonder what kind of relation there is between these quantities, but sadly I could not find any theoretical descriptions of this relation.

The photonic crystal is shown below. It is a rectangular grid of metallic poles in a vacuum. The desired propagation direction is along the z-axis. Also a few of the dispersion lines are shown below, as well as their derivatives (the group velocity).

Finding an analytical description of this relationship seems unlikely to me, but I would like to know what function would describe this lineshape, as this would help with my numerical analysis.

Dispersion diagram The Photonic Crystal

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First of all, this is NOT a 2D photonic crystal since you send light in the $z$-direction (you are using all 3 dimensions) and if the rod is not infinitely long.

I'm not aware of an analytic formula, but your band diagram exhibits several features characteristic of metallo-dielectric photonic crystals. For example, there's a bandgap starting at zero frequency. I suspect that the lineshape is partly due to the finiteness of the rod length.

The mathematics of infinite rods is close to that of photonic crystal fibers at the large-$k_z$ limit (linearly-polarized limit). There are many references on this topic.

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