We consider the problem of the electromagnetic modes which exist in a doubly periodic structure consisting of a square or hexagonal array of cylinders, and focus on a special type of mode called the TEM mode. We discuss the link between the TEM mode, the off-axis mode which tends towards it for small wave numbers and near-normal incidence, and the effective refractive index of the array. For arrays of finitely conducting cylinders we show that the quasistatic limits of TE and TM modes are different. When the structure contains perfectly conducting cylinders, we show that there is no such limit for TM modes, as the corresponding photonic band diagram does not contain an acoustic band. Our results highlight the restrictions regarding the validity of homogenization procedures for doubly periodic composites.