You may need to refer the reference for more details.
I guess that, the vertical axis is equivalent to frequency, and the horizontal axis is equivalent to effective refractive index of the modes. If you can somehow get the effective index, you can get the actual propagation constant beta=k0*neff, where k0 is the wavenumber in vacuum. then you can tranform it to the Block wave vector.
To get the mode effective index, you can use the formula from Sneider and Love's book, which gives the expression of neff as a function of the mode fields.
Or you can use FDE to find the neff for this 2D cross section cavity as an infinitely long waveguide.
Those are what I can think of and suggest , since there is no exmaple on Lumerical website.
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