Title:

Microwave response of finite periodic metal structures

The development of the concept of metasurfaces, the twodimensional version of metamaterials, has unveiled a new paradigm in the control of electromagnetic wave propagation. These consist of periodic arrangements of small scatterers, whose response is engineered locally. The analysis of large but finite arrays is a substantial theoretical and computational problem, and therefore the application of metasurfaces has largely been limited to cases in which the effects introduced by the truncations from the infinitely periodic structures necessary in real applications are expected to be negligible. This thesis focuses on the study of the perturbations of the microwave response of large finite arrays introduced by truncations. To do so very efficient analysis methods based on the method of moments are developed and implemented for different types of truncated slot/patch arrays. The physical insight gained from the analysis of the microwave response of truncated arrays will help in the engineering of new types of metasurfaces that take full advantage of the knowledge provided in here on the different mechanisms governing the coupling between surface waves and free space radiation. In the first part of the thesis, the microwave response of infinite twodimensional and onedimensional periodic arrays of slots in negligiblethickness and perfectlyconducting screens is studied using a specialised implementation of the method of moments whose matrix coefficients are efficiently obtained in the spatial domain. This is done thanks to the analytical calculation of the crosscorrelation between the basis functions used to expand the unknown electric field distribution in the holes and the use of the Ewald's method to obtain rapidly converging expressions for the periodic Green's functions involved. By studying the power transmission coefficient under several symmetries, the effects that these introduce into the appearance of the numericallychallenging phenomenon of extraordinary transmission is studied. This phenomenon has a deep connection to the existence of infinite periodicity in the plane of the array, and provides with a good benchmark of the convergence of the finite array solution to the response of the twodimensional infinite periodic array. It is found that symmetries, dictated both by the disposition of the scatterers with respect to the lattice vectors and by the illumination utilised to excite the array, play an important role in the convergence of the transmission properties of a finite number of periodic rows to the solution of the twodimensional infinite periodic array. By extending the analysis methods to rectangular finite arrays of slots, it is found that the global response of the array, studied in terms of the transmission coefficient through the array, is modified when truncations are introduced. In addition, the local response of the elements of the array, even when these are located in central positions within arrays with large number of elements per row, is altered. It is found that the existence of truncations leads to the excitation of longwavelength surface waves supported by extraordinary transmitting arrays, which under certain conditions can propagate tens of unit cells setting up standing wave patterns on the surface of the array. It is shown both theoretically and experimentally that these surface waves can be used to design very compact extraordinary transmission arrays when one engineers the coupling between the illumination and such waves, in contrast to the very large arrays thought to be required before. In addition, it is also shown that by modifying the relative size of the slots with respect to the periodicity, one can bind these waves to the surface of the array, finding both theoretically and experimentally that in the case of finite arrays one can excite a larger number of surface wave species than was predicted by the dispersion analysis of twodimensional infinite periodic slot arrays. The study of the effects introduced by truncations on the scattering by finite slot/patch arrays is also extended by presenting the rigorous solution of the scattering by a semiinfinite twodimensional array of narrow patches (the complementary problem to that of slots in perfectly conducting screens). It is shown that the currents on the dipoles come from the sum of three wave species: the first one arising from the solution of the infinite array, the second produced by the fields diffracted by the edge of the array that present a continuous kspace spectrum (in contrast to the discrete kspace spectrum imposed by strict infinite periodicities) and the third one comprising the whole set of surface waves supported by the array (which are excited by the diffracted fields as these can not be directly excited by plane waves otherwise). By also studying the spatial distribution of the fields diffracted by the array further insight is obtained into the physical and mathematical differences in the response of finite arrays, leading to analytical formulas that could serve as approximate recipes for the analysis of the scattering by arrays comprising thousands of elements. In the last part of the thesis, the implications that symmetries have on the dispersion characteristics of bound surface waves are studied. In particular, glidesymmetric configurations of periodically notched slots are explored to give physical insight on the inherent properties of higher symmetries (a class to which glide symmetry belongs), such as the nonzero group velocity found at the Brillouin zone boundary, which leads to nearly constant effective refractive index for the surface wave over a large frequency range. Further, it is shown how that special property can be engineered for metasurfaces presenting a simple mirror symmetry. This could lead to metasurfaces with enhanced functionality arising from the easier control of the modes excited on the metasurface, thanks to the ease of selectivity in the excitation of modes with even/odd field distributions. In the context of antenna engineering, the little frequency dependence of the mode index is connected to that of the angle at which the surface waves are radiated when the metasurface is placed near a highindex material, and can lead to broadband leakywave antennas that do not suffer from angle scanning.
