. . . Velocity:¹

You'll get that the group velocity, ${\displaystyle v_g = {c \over 1+{q^2N \over 2m\epsz} \sum_j {(\omega_j^2 + \omega^2)f_j \over \left(\omega_j^2 + \omega^2\right)^2 } } }$

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. . . Modes:²

You'll find that only 4 TE modes will propagate.

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. . . Modes:³

You will deduce that $E_z = E_0 \sin {m \pi x \over a} \sin {n \pi x \over b}$ with m,n integers ( $1,2,3,4,\cdots$) and you will find that the cutoff frequency is $\omega_{mn} = c \pi \sqrt{ \left(m \over a\right)^2 + \left(n \over b\right)^2 }$, the same as for TE modes.

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