This thesis focuses on the understanding and computation of the dispersion properties of periodic structures possessing higher symmetries with the multimodal transfer matrix method (MMTMM). Periodic structures with higher symmetries are invariant after additional symmetry operations over the translation operation. To demonstrate the potential of the MMTMM, three structures with spatial higher symmetries are proposed and their operation explained based on the constituent modes. 

In this thesis, I propose, analyze and explain the operation of two structures possessing glide symmetry and one with twist symmetry. Glide-symmetric structures remain invariant after a mirroring and a translation whereas twist-symmetric structures remain invariant after n rotations and translations. These structures inherently have low dispersion due to the interactions of the fundamental mode with higher order modes.

The MMTMM has been implemented in order to efficiently compute the complex propagation constant of these structures. This is a hybrid method that models a unit cell as a multiport network. Each port accounts for one mode, so the coupling between modes is considered. Commercial software is used to compute the ABCD-matrix, then post-processing is used to get both the phase and attenuation constant due to material losses, electromagnetic bandgaps and/or radiation. This method permits the study of complex structures while enabling a fundamental understanding of the modes that contribute to the dispersion properties, as well as their interactions. 

The first periodic structure analyzed in this thesis is a dielectric-filled corrugated waveguide. It allows the propagation of a backward mode in a wide frequency band. A discussion on the convergence of the method concludes that it is needed families of TE/TM modes with the same number of variations in the x direction. 

The second structure is a glide-symmetric dielectric unit cell placed in a parallel plate waveguide. This unit cell can be used to produce planar lens antennas that can be cost-effectively manufactured with dielectric 3D-printers. The attenuation constant due to material losses in two different directions is computed using the MMTMM.

Finally, a 3-fold twist-symmetric dielectric open waveguide is analyzed. Its interest lies in its inherent circular polarization selectivity. Here, the MMTMM is used to compute the attenuation constant from material losses and the stopband, as well as to understand the interaction between linear and circularly polarized modes.