**Project objective and realized outcomes**

Curved multilayered dielectric structures with embedded metallic patterns (commonly referred to as metasurfaces) can act as electromagnetic devices that direct waves, manipulate the polarization of transmitted or reflected waves, or influence the spectrum properties of those waves. Recently, it was shown that these properties of metasurfaces can be successfully used for building various electromagnetic structures applied in different applications that vary from smart radomes for airplane or missile applications (in particular nose radomes) to subreflectors for multi-frequency antenna reflector systems. Till now most of the attention was focused on the development of planar metasurface structures, however, many demanding electromagnetic applications require the implementation of curved structures. Introduction of curvature significantly complicates the analysis and the design since the quasi-infinite periodicity used in planar structures is lost in this case (if the structure is curved in both principal directions). Furthermore, the considered structures are very large in terms of wavelength and they contain a lot of small metallic details within each of the metasurface layers (by definition, the unit cell of the metasurface pattern is much smaller than the wavelength). With all this in mind in order to successfully design curved metasurface structures one needs to develop a specialized efficient numerical analysis algorithms. The need for this specialized program lies in the fact that these large finite structures with numerous small cells cannot be efficiently designed using general electromagnetic solvers since the needed computer time for analyzing such complex structures would be very long (quite often more than several tens of hours or even days), memory requirements would be extremely large and the successful optimization would simply be too slow.

The objective of the project is to develop a systematic approach and associated computer programs for designing curved multilayer structures containing metasurface layers. First part of the project was focused on the development of an efficient and accurate program for designing such structures, while second part was focused on the development of experimental prototypes that demonstrated the abilities of the developed analysis and design approach.

The development of efficient and accurate programs was based on combining several approaches that show superior properties when designing electromagnetic structures containing metasurface layers:

**(1) Modeling of a curved metasurface sheet**

Metasurface structures are usually very complex and it is extremely time consuming to directly analyze them using general EM solver (note that it is practically impossible to use optimization procedure in such cases since one evaluation of the cost function takes typically numerous hours). Therefore, one needs good approximate models of metasurface sheets without going into details about their geometries. In the planar case the metasurface sheets were successfully analyzed using the surface sheet impedance approach. Therefore, as a first step we have developed a model of curved metasurface sheets using the modified surface impedance approach. In planar case the value of surface impedance tensor Z depends on frequency, polarization and angle of incidence. In conformal case we need to include principle radii of curvature and the EM wave variation along the metasurface of the impinging wave. Both of them play a crucial role in selecting types of elements for building curved metasurface sheets. Note that the local value of the surface impedance tensor Z can be determined using either the specialized software, approximate formulas, or the general electromagnetic software.

**(2) Analysis of canonical curved multilayer metasurface structures**

The second step in developing general program for analyzing curved metasurface structures was the development of a program for analyzing canonical curved multilayer metasurface structures – cylindrical and spherical ones. There are two main reasons why we have chosen this intermediate step. The first reason is that the program for the analysis of canonical metasurface structures is extremely fast and thus the needed computer time for getting the parameters of the considered design is very short. Therefore, we have connected the developed analysis programs with a global optimization routine in order to obtain the initial design of the metasurface structure, which can serve as a starting point for making a design of the final general structure. Another reason for developing the program for analyzing canonical structures, perhaps even more important, is that such program makes it possible to investigate all the effects that are caused by the bending of metasurfaces. In other words, by analyzing canonical structures it is possible to understand what are the differences in the electromagnetic parameters of curved metasurfaces (compared to the planar structures) and to incorporate this knowledge into the process for designing general curved metasurface structures.

The analysis of canonical multilayer metasurface structures is based on combining the surface impedance approach for modeling the curved metasurface sheet with the ABCD matrix approach for modeling cascaded metasurface structures. We have made the extension of ABCD matrix approach (originally derived for planar multilayer structures) to include the cylindrical and spherical geometries. The generalization is based on representing the curved dielectric layer as cylindrical or spherical transmission lines, and proper description of transmission line modes was used in the derivation of ABCD matrix parameters.

The surface impedance tensor

To conclude, the reported method for analysis of canonical curved multilayer metasurface structures represents a powerful starting point in the design of general curved metasurface structures. Furthermore, the proposed method aids in selecting the type of patterned sheet that is needed to obtain a desired angular variation of surface impedance tensor

**(3) Analysis of curved metasurfaces with spatially-varying impedance distribution**

Some devices like e.g. cylindrical or spherical cloaks would require homogeneous distribution of surface impedance. However, many devices would require spatially-varying impedance distribution in order to modulate amplitude and phase of incoming wave. Furthermore, quite often the considered metasurface is occupying only a part of canonical surface (such as a cylinder or a sphere). One example is a dome antenna with a purpose to either flatten the gain dependency or to enhance the gain of the antenna array inside the dome. Consequently, the analysis approach for canonical curved structures was generalized in order to include a class of metasurface structures with spatially-varying impedance distribution. There are several ways how to make this generalization. As demonstrated in the report, the most suitable selection depends on the domain in which we would like to calculate the metasurface sheet impedance distribution (i.e. spatial or spectral domain) and on the way how we would like to calculate the needed Green’s functions. Fortunately, all the possibilities lead to a solution of the same complexity and accuracy.

**(4) Analysis of general curved multilayer metasurface structures with Body-of-Revolution (BoR) type of symmetry**

Most curved metasurface structures of interest (smart radomes, nose radomes, reflectors and subreflectors, etc.) contain Body-of-Revolution (BoR) symmetry. Therefore, in order to develop a fast and accurate program for design purposes, we have decided to develop a computer program for curved metasurfaces structures with BoR type of symmetry. By this a general three-dimensional electromagnetic problem is transformed into a spectrum of two-dimensional problems which are much easier and faster to solve. The developed program is very general, it is based on the Moment Method with sub-domain basis and test functions (used to solve two-dimensional problems), the analyzed structure can be multilayer and can contain arbitrary number of metasurface layers, the value of the surface impedance tensor can vary spatially and can have different values for different angular variation of the incident wave, etc. The developed program is written in a form of an algorithm. We named the algorithm the G2DMULT-BoR algorithm, i.e. the algorithm that calculates the Green’s functions of two-dimensional (2D) multilayer structures with body-of-revolution type of symmetry.

**(5) Experimental verification of the developed analysis and design method**

Second part of the project was focused on the realization of practical curved metasurface prototypes needed to answer two questions – is it possible to successfully use the developed computer programs for designing realistic electromagnetic devices, and which technology can be used for making such structures. The planar metasurface structures can be easily built using the printed circuit board (PCB) technology. The same technology can be used for singly-curved structures. One need to consider the flexural modulus of selected substrate, i.e. one has to select a substrate that is suitable for bending (luckily, the substrate producers offer such products). However, double-curved structures cannot be built in such a way and some alternative technology should be implemented. In recent years, there is an expansion of 3D printing methods and we propose to test if such technology can be used for building double-curved metasurface structures. The main problem in this realization will be to obtain accurate metallization of the desired pattern, and as a part of the project we have investigated several metallization approaches. The easiest approach is to use a spray with conductive (most often silver-based) paint. However, such method is not suitable for obtaining metasurfaces with geometrically small details. Furthermore, the ohmic lossess are larger comparing with the structures produced using printed circuit technology. Much better properties can be obtained with sputtering process (however, one needs to have access to such production technology). The last considered option is metallization of 3D printed structures using electroless copper plating process. It is important to note that in all considered methods one first needs to manufacture or print an appropriate mold pattern or template to apply the metallization on the dielectric sample.