Recently, innovative adaptations of the Ritz method incorporating deep learning have been developed, known as the Deep Ritz Method. This approach employs a neural network as the trial function for variational problems. However, the neural network does not inherently satisfy the boundary conditions of the variational problem. To address this issue, the Deep Ritz Method introduces a penalty term into the functional, which is strongly dependent on hyperparameters and may lead to misleading results during the optimization process. In this work, we propose an ansatz that inherently satisfies the boundary conditions of the variational problem, thereby eliminating the need for penalty terms. A key contribution of this study is that all supporting theorems and corollaries are established in Sobolev norms, which constitute the natural framework for variational problems, as the functional depends explicitly on the solution and its derivatives. This provides a rigorous justification for the expressiveness and admissibility of the proposed ansatz within the Ritz method. The results demonstrate that the proposed ansatz not only avoids misleading optimization outcomes but also reduces complexity while maintaining accuracy, highlighting its practical effectiveness for solving variational problems.

A generalization of the efficient interpolation of periodic Green’s functions is presented for a multilayer medium hosting transverse electric current densities and transverse equivalent magnetic current densities at different interfaces. The mathematical model is realized in terms of Maxwell’s equations for multilayer media with isolated electric and magnetic equivalent current densities for large values of spectral variables or small values of spatial variables. This fact enables the use of Mixed Potential Integral Equation (MPIE) approaches in the spectral domain and provides asymptotic behaviors for Green’s functions of vector and scalar potentials for both electric and magnetic sources. Consequently, the singular behaviors of the Green’s functions around the source point are obtained as the spatial counterpart of the proposed spectral asymptotic behaviors. Thus, regularized multilayer periodic Green’s functions are obtained, which can be efficiently interpolated over the entire unit cell using Chebyshev’s polynomials.

Satellite reflectarray synthesis using the intersection approach by means of algorithms of the alternating generalized projection (AGP) method to avoid trap points with sequential stages of different illuminations is presented. In this scenario, we examine two distinct sets: the set of radiation patterns producible by the reflectarray and the set of radiation patterns which meet the mission’s criteria. These sets are generally non-convex. Therefore, it is expected that conventional algorithms of the method of alternating projections (MAP) converge to trap points (i.e., local minima of the distance between the involved sets). Thus, the AGP method, which takes into account the trap points, has been considered. When large reflectarrays are considered for satellite applications, several trap points can appear, producing oscillatory behavior between several trap points. A mathematical formalism which supports the idea that a reduction in the edge illumination of an antenna involves fewer trap points is described. Thus, the oscillatory behavior can be avoided by using sequential stages in the synthesis process with different edge illumination levels in each stage. In this work, we demonstrate this synthesis technique with sequential stages, using the proposed algorithm to avoid trap points.

Promising algorithm of alternating generalized projection method (AGP) is proposed for phase-only synthesis process of satellite reflectarray antennas under intersection approach. This promising algorithm is a hybridized algorithm of two specialized algorithms for non-convex sets rescued in the literature: algorithm based on separating hyperplanes and algorithm based on decomposition method in polar cones. Since the sets involved in phase-only synthesis process of satellite reflectarray antennas are non-convex, the conventional von Neumann alternating projection method proposed in the literature does not guarantee convergence to the point in the intersection of the involved sets. In addition, the results of the phase-only synthesis obtained by the different algorithms were compared and promising improvements produced by the proposed hybridized algorithm were shown.

In this work, a reflectarray antenna is presented as a dual-polarized plane wave generator (PWG) in mm-wave frequencies. The reflectarray produces a uniform plane wave within its near field in a volume close to the aperture. A dual-polarized pencil beam reflectarray is used as starting point to carry out a phase-only synthesis (POS) to reach a dual-polarized uniform plane wave. The synthesis aims to overcome the amplitude limitation due to the incident field while preserving the phase flatness. The phase distribution obtained in the POS is used to carry out a design based on sets of orthogonal-coplanar dipoles. This unit cell provides independent control of each polarization using a single-layer topology. The design is manufactured, and the prototype is measured in a planar acquisition range facility. The prototype shows discrepancies with simulation and further analysis is carried out. After finding out the proper value of the dielectric constant, a new design is carried out. The second prototype shows a good agreement with simulations, obtaining that reflectarray antennas are a suitable and low-profile solution to produce uniform plane waves within the near field.

Fast computation of the coefficients of the reduced impedance matrix of the method of moment (MM) is proposed by expanding the basis functions (BFs) in pulses and solving an equivalent periodic problem (EPP) for analyzing large multilayer structures with non-uniform rational basis spline (NURBS) modeling of the embedded layout. These coefficients are required by the computation of sparse approximate inverse (SAI) preconditioner, which leads an efficient iterative version of the MM. This reduced coefficient matrix only considers the near field part of the MM matrix. Discrete functions of small sizes are required to implement the pulse expansion and EPP. These discrete functions of small size lead to discrete cyclic convolutions that are computed in a very fast way by fast Fourier transform (FFT)-accelerated matrix–vector multiplication. Results obtained using a conventional laptop show an analysis of very large multilayer structures with resonant layouts, as whole reflectarrays of electrical size 40 times the vacuum wavelengths, where the iterative MM with a SAI preconditioner can be 22.7 times faster than the pure iterative MM without any preconditioner.

A comparison between Ma-Rokhlin-Wandzura (MRW) and double exponential (DE) quadrature rules for numerical integration of method of moments (MoM) matrix entries with singular behavior is presented for multilayer periodic structures. Non Uniform Rational B-Splines (NURBS) modelling of the layout surfaces is implemented to provide high-order description of the geometry. The comparison is carried out in order to show that quadrature rule is more suitable for MoM matrix computation in terms of sampling, accuracy of computation of MoM matrix, and CPU time consumption. The comparison of CPU time consumption shows that the numerical integration with MRW samples is roughly 15 times faster than that numerical integration using DE samples for results with similar accuracies. These promising results encourage to carry out a comparison with results obtained in previous works where a specialized approach for the specific analysis of split rings geometries was carried out. This previous approach uses spectral MoM version with specific entire domain basis function with edge singularities defined on split ring geometry. Thus, the previous approach provides accurate results with low CPU time consumption to be compared. The comparison shows that CPU time consumption obtained by MRW samples is similar to the CPU time consumption required by the previous work of specific analysis of split rings geometries. The fact that similar CPU time consumptions are obtained by MRW quadrature rules for modelling of general planar geometries and by the specialized approach for split ring geometry provides an assessment for the usage of the MRW quadrature rules and NURBS modelling. This fact provides an efficient tool for analysis of reflectarray elements with general planar layout geometries, which is suitable for reflectarray designs under local periodicity assumption where a huge number of periodic multilayer structures have to be analyzed.

In this work, a reflectarray is proposed to be used as a probe of reduced and portable compact-antenna-test range for 5G new radio devices. The reflectarray works at 28 GHz and produces a quiet zone in the near-field region of the antenna. Considering that the quiet zone specifications are established in terms of the amplitude and phase ripple, a synthesis technique is presented to optimize the near field in the Fresnel region of a reflectarray with amplitude and phase constraints. The proposed technique is based on the generalized intersection approach using the Levenberg-Marquardt algorithm as its backward projector obtaining a novel technique in near-field synthesis. This technique is applied to improve the quiet zone radiated by the proposed reflectarray, overcoming the amplitude and phase limitations of the initial configuration. The solution provided by this process is used to design and manufacture a reflectarray based on a three-parallel-dipole cell. Finally, the prototype is measured in a near-field planar range facility in order to evaluate the radiated quiet zone and demonstrate the methodology. The prototype satisfies the tight requirements in amplitude and phase, obtaining promising results.

In this work circularly polarized flat reflectarray antennas are designed to work at downlink frequencies in Ku- and Ka-band. The design approach for circular polarization radiation combines both the variable rotation technique (VRT) and the element size variation technique (ESVT). The reflectarray element employed consists of a rotable split ring operating at 12 GHz (Ku-band) that surrounds two orthogonal sets of parallel dipoles of variable size operating at 20 GHz (Ka-band). Rotations of split rings and different sizes of dipoles are exploited as degrees of freedom by VRT and ESVT for designing focused beams dual band dual circular polarized reflectarrays. The split ring can be used to achieve single focused beams for circular polarization by VRT at 12 GHz, and the set of orthogonal dipoles can be used to achieve two focused beams for dual circular polarization by ESVT at 20 GHz. In this way, the reflectarray element has been used to design two flat dual band circularly polarized reflectarrays that generate three pencil beams either in the same direction or in different directions: one pencil beam with right-hand circular polarization (RHCP) at 12 GHz and two pencil beams with both RHCP and left-hand circular polarization (LHCP) at 20 GHz. The designed reflectarrays are carried out under local periodicity assumption by means of in-house electromagnetic software that applies the Method of Moments in the Spectral Domain. Validations of the in-house electromagnetic software show significant CPU time savings with respect to CPU time consumption provided by commercial software of general purpose as CST electromagnetics software. Numerical results of radiation patterns produced by the designed reflectarrays show efficiencies around 65% and a bandwidth of 6% for a main beam cross-polarization discrimination of 25 dB. The numerical results are satisfactorily validated by CST software and a tolerance errors study has been numerically carried out with acceptable results.

This contribution describes the design of a reflectarray to produce two adjacent beams per feed in orthogonal circular polarization (CP) simultaneously at transmit (Tx) and receive (Rx) frequencies in the Ka band. The discrimination in CP is achieved by the variable rotation technique (VRT), whose dual band operation is validated for the first time by the manufacture and measurement of a 25 cm $\times \,\, 25$ cm reflectarray demonstrator. Moreover, an optimization procedure has been implemented to improve the narrowband performance of the VRT (which leads to high cross-polar levels when the VRT is applied). The reflectarray prototype has been designed to deviate by ±5° the beams radiated by a dual-CP feed-horn at Tx and Rx frequencies in the $Ka$ band (20/30 GHz). The results are satisfactory and validate the concept of generating two spaced beams in orthogonal CP and dual band by the VRT with a single feed. This concept can be suitable for multiple spot satellites in the $Ka$ band, enabling a reduction in the number of antennas and feeds needed to provide the multispot coverage.

BICGSTAB-FFT method of moment (MM) scheme is proposed to analyze several levels of planar generic layouts embedded in large multilayer structures when the layout geometries are modeled by NURBS surfaces. In this scheme, efficient computation of normalized error defined in iterative bi-conjugate gradient stabilized (BICGSTAB) method for large multilayer structure analysis problems is implemented. The efficient computation is based on pulse expansion with dense equi-spaced mesh of generalized rooftop basis functions (BFs) defined on NURBS surfaces and equivalent periodic problem (EPP) in order to apply fast Fourier transforms (FFT). Moreover, efficient computation of Green's functions for multilayer structure is implemented for near and far field regions. Experimental and numerical validations of whole printed reflect array antennas of electrical size between 8 and 16 times the vacuum wavelengths are shown. In these validations, CPU time consumptions of the proposed method are obtained with results between few minutes and half an hour using a conventional laptop.

The bandwidth behavior has been studied and improved for a reflectarray cell formed by two symmetric arcs and dipoles printed on two layers, which uses Variable Rotation Technique at two frequencies for dual circular polarization. First, the appropriate thickness of the dielectric layers have been selected to improve the bandwidth. Then, an optimization routine has been applied to minimize the phase errors in a frequency band from 29.25-29.75 GHz. As a result of this optimization, the phase errors have been drastically reduced from 40 to 3 degree.

In this paper, an efficient technique of computation of method of moments (MM) matrix entries for multilayer periodic structures with NURBS surface and Bézier patches modelling is proposed. An approximation in terms of constant pulses of generalized rooftop basis functions (BFs) defined on Bézier patches is proposed. This approximation leads discrete convolutions instead of usual continuous convolution between Green’s functions and BFs obtained by the direct mixed potential integral equation (MPIE) approach. An equivalent periodic problem (EPP) which contains the original problem is proposed to transform the discrete convolutions in discrete cyclic convolutions. The resultant discrete cyclic convolutions are computed by efficiently using the Fast Fourier Transform (FFT) procedure. The performance of the proposed method and direct computation of the MM entries are compared for phases of reflection coefficient. The proposed method is between 9 and 50 times faster than the direct computation for phase errors less than 1 deg. The proposed method exhibits a behaviour of CPU time consumption of O(NbLog10Nb) as the number Nb of BFs increases. This behaviour provides significant CPU time savings with respect to the expected behaviour of O(Nb2) provided by the direct computation of the MM matrix entries.

Multi-beams antennas are currently being used for direct broadcast satellite, personal communication satellite, military communication satellite, and high-speed internet applications. In this work, a circularly polarized (CP) multi-spot beam satellite parabolic reflectarray antenna is designed to provide six spot beams at 19.7 GHz. For this purpose, an easy technique to compute the required phase shifts to produce two focused beams in specular directions for a CP parabolic reflectarray is proposed. These required phase shifts are added to the reflected fields by the variable rotation of the reflectarray elements printed on the surface of a parabolic antenna which are fed by a dual-CP feed-horn. For this purpose, a reflectarray cell made of a conductive cross embedded in a grounded multilayered substrate is optimized to produce very linear phase-shift and low cross-polarization level. To demonstrate the multibeam capacity, a 1.8-meter offset parabolic reflectarray made of the optimized reflectarray element was designed to generate six focused beams in dual-CP with three dual-CP feed-horns. The six main spots fulfill the typical multi spot satellite requirement with angular separation less than 0.56°, 0.4 dB loss in the gain, and cross-polarization level below 35 dB with respect to the maximum of radiation.

One planar 18 x 18 cm(2) reflectarray prototype, capable of generating adjacent pencil beams with the orthogonal circular polarizations (CPs) in the Ka-band (19.2-20.2 GHz), has been designed, manufactured, and measured as a proof-of-concept to demonstrate the possibility of beam separation in the dual-CP applications. The prototype exhibits main beams pointing at elevation angles of 16.7 degrees and 21.3 degrees for left-hand CP and right-hand CP, respectively. A 1.5 dB gain variation and 66% of efficiency are achieved in the frequency band 19.2-20.2 GHz for a design center frequency of 19.7 GHz. These results will be useful for the future work, involving the design of multispot dual-CP and dual-band planar reflectarrays.

A dual-frequency reflectarray cell is proposed in this work to provide a progressive phase shift of opposite sign in each circular polarization (CP), by applying a variable rotation technique (VRT) [1]. The proposed reflectarray cell consists of two symmetrical arcs printed on the top surface and two orthogonal sets of parallel dipoles printed on two levels of a grounded dielectric substrate, see Figure 1. The arcs and dipoles are rotated independently to control the phase in CP at the lower (19.7 GHz) and higher (29.5 GHz) frequencies respectively. This reflectarray cell can be used to generate two adjacent beams in orthogonal CP per feed at transmission (Tx) and reception (Rx), in order to reduce the number of feeds and antennas required in current multi-spot satellites operating in Ka-band, as proposed in [1] for a single frequency band.

A new reflectarray cell is proposed to simultaneously provide opposite phase shifts between orthogonal circular polarizations at two frequencies (19.7 and 29.5 GHz) by applying a variable rotation technique independently at each frequency. The reflectarray cell, which consists of two dielectric layers with two levels of printed elements (dipoles and arcs), has been characterized for both circular and linear polarizations at each frequency. Apart from the implementation of the variable rotation technique at each frequency, the reflectarray cell provides an additional phase adjustment at the higher frequency, which can be used to shape the beam. This concept applied to multiple spot satellites in Ka-band will allow to produce two adjacent beams per feed in orthogonal circular polarizations at transmission and reception frequency bands.

The concept of generating two spaced beams in dual-CP by variable rotation technique (VRT) is validated for the first time by manufacturing and measuring. A reflectarray demonstrator has been designed, manufactured, and tested to deviate +/- 10 degrees the beam of a dual-CP multiflared horn at 19.7 GHz. The proposed reflectarray cell for VRT is made of a conductive cross printed on a grounded dielectric. The experimental results show that two adjacent beams in dual-CP maintain the correct directions in the prescribed frequency band (19.2-20.2 GHz). The results are satisfactory and validate the concept for generating two spaced beams in orthogonal CP by VRT with a single feed.

A design technique has been proposed to generate two closely spaced beams in orthogonal circular polarization using the variable rotation technique (VRT) in reflectarrays printed on arbitrary shaped surfaces fed by a single dual circular polarized (CP) horn. The proposed reflectarray cell for VRT is made of a conductive cross printed on a grounded dielectric. A 1.8-meter offset parabolic reflectarray has been designed to generate six beams in circular polarization with three dual-CP feed-horns, with very promising results for multi spot beam satellite antennas in Ka-band.

This contribution describes a design concept of a reflectarray antenna to produce four adjacent beams per feed through the simultaneous use of polarization and frequency discrimination. The feed position is computed to produce two adjacent beams in different frequencies accounting for the beam squint effect, which ensures a minimum phase variation between the phase distributions at the two frequencies. The other two beams in orthogonal polarization are generated by implementing in the reflectarray a different phase shift for each polarization. This contribution presents the design, manufacturing, and measurement of a 43 cm demonstrator that operates at transmit frequencies in the Ka-band. The proposed concept can be suitable for multiple spot-beam satellites in the Ka-band, enabling a reduction in the number of antennas and feeds needed to provide the multispot coverage.