Ultra-thin metamaterials, called meta-surfaces or meta-sheets, open up new opportunities in designing microwave radomes, including an improved transmission over a broader range of antenna scan angles, tailorable and reconfigurable frequency bands, polarization transformations, one-way transmission and switching ability. The smallness of the unit cells combined with the large electrical size of microwave radomes significantly complicates full-wave numerical simulations as a very fine sampling over an electrically large area is required. Physical optics (PO) can be used to approximately describe transmission through the radome in terms of the homogenized transmission coefficient of the radome wall. This paper presents the results of numerical simulations of electromagnetic transmission through planar meta-sheets (infinite and circularly shaped) obtained by using a full-wave electromagnetic field simulator and a PO-based solution.

Microwave antennas must be protected against external mechanical (aerodynamic drag, wind) and chemical (ice, rain) effects. Consequently, protective coverings are needed, which are called radomes (radar domes). The shape of the radome must be typically conformal with the shape of the platform (car, aircraft, ship). An ideal microwave radome must be fully transparent at microwave frequencies, at least over the bandwidth of the enclosed antenna. However, losses in the power and distortions in the phase of transmitted/received signals caused by realistic radome walls may lead to significant degradation in antenna performance.

The most noticeable effect is the reduction of the transmitted power, which leads to a reduced range of a radar device. A further effect is the growth of the received echo. Reflections, eventually multiple reflections, within the volume enclosed by the radome are the reason for the existence of standing waves in the transmission line of the system, which alters the magnetron frequency (“pulling”) and is a reason for the loss of signals. Pulling depends on the magnitude and phase of the standing waves and therefore on the reflection coefficient of the radome wall. Furthermore, refraction in the walls of the radome and phase distortions upon transmission through the radome wall may result in an apparent shift in the angular position of targets (Cady et al., 1948).

If an improved transmission is required at a specific frequency or in a limited frequency band, meta-sheets can be utilized to design the radome walls. Meta-sheets (also metasurfaces, metafilms) are quasi two-dimensional metamaterials that consist of a single layer of electrically small resonators (inclusions) embedded in an electrically thin layer of a dielectric material (substrate). Meta-sheets allow a wide range of possibilities in controlling transmission, reflection and absorption of electromagnetic waves, e.g. Tretyakov (2015). Similarly to metamaterials (Capolino, 2009), meta-sheets are in general periodic arrays of subwavelength unit cells placed in a substrate. Since the size of the unit cell is much smaller than the wavelength of the incident wave, meta-sheets appear to be effectively homogeneous with the effective material parameters which are much broader than in natural materials and can be adjusted to achieve a desired electromagnetic behaviour of the sheet. Meta-sheets can bring new features to microwave radomes such as tailorable and reconfigurable frequency bands, polarization transformations, one-way transmission and switching ability (Öziş et al., 2015, 2016).

Typically, radomes are electrically large structures and, therefore, simulation of transmission and reflection properties of a radome with full-wave numerical methods (FEM, FDTD) can be a difficult, memory- and time-consuming task. An additional complication, specific for meta-sheets and metamaterials, is the need for extremely fine spatial discretization, which results in a very large number of unknowns in the discretized problem. Electromagnetic scattering from electrically large objects may well be calculated by high-frequency approximations based on physical optics (PO), e.g. Albani et al. (2011), Obelleiro-Basteiro et al. (1995), Youssef (1989). The PO equivalent currents induced by an incident wave on a meta-sheet portion of the radome can be calculated from reflection and transmission coefficients of the locally conformal planar infinite meta-sheet (tangent plane approximation). The transmission and reflection coefficients of planar infinite meta-sheets can be relatively easily calculated since the simulation volume can be reduced to a single unit cell of the meta-sheet by using periodicity of the structure. So, PO combined with numerically obtained transmission and reflection coefficients of the tangent meta-sheets can be a good alternative to full-wave methods, but applicability of the results should be validated through comparisons with full-wave simulations.

In this paper, PO is applied to electrically large planar circular
meta-sheets to calculate the transmitted electric field. Isotropic
ring-shaped metal particles are used as inclusions to make the structure
polarization-independent. The high frequency structure simulator (HFSS)
(ANSYS, 2016) is used to calculate the effective transmission coefficient of
the meta-sheet (to be employed in the framework of the PO approach) and to
perform full-wave simulations of the transmitted field in the presence of an
electrically large meta-sheet of finite size. The PO and full-wave
electromagnetic field simulation results will be compared on the optical axis
of the configuration, where the PO solution can be expressed in a
particularly simple form and the error of the PO solution is expected to be
at maximum. The operation frequency is chosen to be at 10

This paper is organized as follows. In Sect. 2, we briefly review the
principles of the PO approximation and derive a PO solution for a circular
partially transparent plate described by a transmission coefficient. The HFSS
setup for full-wave simulations of transmission through an infinite planar
meta-sheet and through a finite electrically large circular meta-sheet is
presented in Sect. 3. In Sect. 4, the PO approach and full-wave
electromagnetic field simulation are applied to an electrically large
perfectly conducting (PEC) disc to show that the chosen structures are
sufficiently large in order that the PO approximation to be valid and then to
penetrable meta-sheets of the same diameter. The conclusions are given in
Sect. 5. The time dependence

An infinite plane

PO approximation is a high-frequency approximation, which in contrast to geometrical optics (GO) accounts for the finite size of the wavelength (Macdonald, 1902; Mentzer, 1955; Osipov and Tretyakov, 2017; Saez de Adana et al., 2011; Ufimtsev, 2014). It can be defined as a combination of GO with the field equivalence principle. According to the equivalence principle the fields scattered by an object can be calculated by integration of equivalent electric and magnetic currents over a surface enclosing the object (Collin, 1960). The equivalent currents are given by tangential electric and magnetic field components on the integration surface. Since the exact fields are a priory unknown, GO approximation for the currents is used, assuming that the integration surface is almost flat on the scale of the wavelength (tangent plane approximation).

Consider an infinite planar semi-transparent screen

To calculate the field at an observation point to the right of the screen in
the PO approximation, we use the infinite plane

Equations (5), (6) and (7) give an exact representation of the field
transmitted through the screen. In the PO approximation,

In this paper, we assume that a part of the screen is the backside of a
homogeneous semi-transparent disc of radius

As full-wave electromagnetic field simulator, we use HFSS which is based on the finite element method (Huebner et al., 2001). The simulator divides the solution domain into a finite number of elements. After converting the entire domain into a discretised domain, a suitable interpolation function is chosen. The interpolation functions can be first, second, or higher order polynomials. The unknown fields are approximated by known basis functions with unknown expansion coefficients. The expansion coefficients are determined by solving a system of linear equations.

Simulation of infinite planar meta-sheets consisting of identical unit cells
can be reduced to a single unit cell with boundary conditions accounting for
periodicity of the structure. In the framework of HFSS, a unit cell as shown
in Fig. 2 is placed in an air-box with master/slave boundary conditions and
Floquet ports for excitation (Fig. 3). The surfaces of the air-box which are
perpendicular to the

The unit cell of a meta-sheet consists of a copper ring embedded in a square piece of a dielectric substrate (FR4 epoxy).

HFSS simulation volume.

The parameters of the unit cells and the corresponding transmission coefficients obtained with HFSS. The outer radius and inner radius describe the shape of the rings.

We have studied transmission through circular planar meta-sheets 160 and
340

The circular meta-sheet in HFSS: electrically small rings are embedded into an electrically large circular FR4 disc.

The electrically large circular PEC disc is of the same size as the meta-sheet in Fig. 4.

Scattered electric field along the observation line. The diameter of
the circular PEC disc is 160

The same as in Fig. 6 but for a PEC disc with the diameter
340

Scattered electric field along the observation line. The diameter of
electrically large meta-sheet is 160

Scattered electric field along the observation line. The diameter of
electrically large meta-sheet is 340

The PO solution given by Eqs. (13), (14) and (17) has been applied to calculating the field scattered by the circular discs (PEC and meta-sheet) and compared with the results of the full-wave simulations. The integral in the PO solution (14) has been calculated by numerical integration using an adaptive integration algorithm, which recursively subdivides the integration region as needed.

The results of simulations for the PEC discs with the diameter 160 and
340

Meta-sheets approach homogeneous materials as the size of the unit cells gets
significantly smaller than the wavelength (

For the disc of 340

For calculation of fields transmitted through finite electrically large meta-sheet structures, a hybrid method using PO approximation with transmission coefficients obtained from full-wave calculation for an infinite planar periodic structure has been proposed.

A comparison of the results of full-wave numerical simulations with the PO predictions for finite planar structures of moderately large electrical size has been performed. The results are in a good agreement as long as, on the one hand, the size of the structure is greater than several wavelengths and, on the other hand, the unit cells are sufficiently small in order to make the sheet effectively homogeneous. As soon as one or both of these conditions are not fulfilled, the PO approach does not apply any more.

Full-wave numerical calculations of meta-sheets are limited to structures of several wavelengths in size. Application to larger meta-sheets is hardly possible because of difficult meshing, large memory consumption and too long computation times. The PO approach discussed in this paper is a good alternative applicable to electrically large structures, planar or curved, provided that the meta-sheet is effectively homogeneous.

The underlaying results of numerical simulations are available from the authors.

The authors declare that they have no conflict of interest. The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association. Edited by: R. Schuhmann Reviewed by: two anonymous referees