TECHNICAL FIELD

The present disclosure relates to a metasurface for electromagnetic waves, an antenna system and a method for designing a metasurface. The present disclosure relates to a computer program and a computer-readable storing medium.

BACKGROUND

Metasurfaces for electromagnetic waves are artificially generated surfaces that modulate the behavior of electromagnetic waves. In this context “artificially generated” is to be understood in the sense that a metasurface is not a surface that already exist in nature. For example, a metasurface may modulate at least one of amplitude, phase and polarization of electromagnetic waves that are reflected by the metasurface when impinging on the metasurface or electromagnetic waves that transmit (i.e. propagate) through the metasurface when impinging on the metasurfaces, depending on the type of metasurface (i.e. whether the metasurface is a reflection metasurface or transmission metasurface). In other words, electromagnetic waves impinging on the metasurface may be reflected by the metasurface in case the metasurface is a reflection metasurface. Electromagnetic waves impinging on the metasurface may transmit (i.e. propagate) through the metasurface in case the metasurface is a transmission metasurface. Optionally, a part of a metasurface may modulate electromagnetic waves impinging on the metasurface by reflecting them and another part of the metasurface may modulate electromagnetic waves impinging on the metasurface by transmitting or allowing the electromagnetic waves through the metasurface. Metasurfaces for electromagnetic waves may also be referred to as “electromagnetic metasurfaces”. Since electromagnetic waves may transmit through the metasurface and/or be reflected by the metasurface, the term “reflection and/or transmission metasurface” may be used as a synonym for the term “electromagnetic metasurface”. Metasurfaces for electromagnetic waves may comprise subwavelength structures (e.g. nanostructures) for modulating the behavior of electromagnetic waves. In the following the terms “metasurface for electromagnetic waves”, “electromagnetic metasurface” and “reflection and/or transmission metasurface” are abbreviated by the term “metasurface”. SUMMARY

A metasurface comprises unit cells, wherein each unit cell of the unit cells has a unique phase response. For instance, in a group of unit cells of the unit cells of the metasurface the unit cells may cover a 2K phase shift, wherein each unit cell of the group of unit cells may have a unique phase response. The terms “phase shift” and “phase change” may be used as synonyms. That is, in a 2K phase shift among unit cells of the metasurface, the phase response of each unit cell of the unit cells covering the 2K phase shift may be unique. The aforementioned group of unit cells (covering a 2K phase shift) may form a super cell of the metasurface, i.e. it may be referred to as “super cell”. In the metasurface the aforementioned group of unit cells, i.e. the super cell, may be repeated. Thus, in two or more super cells of the metasurface there may be corresponding unit cells that have a same unique phase response. The repetition of the super cell, i.e. of the group of unit cells covering a 2K phase shift, may be periodic. For determining the effect of the metasurface on electromagnetic waves impinging on the metasurface the unit cells of the metasurface may be designed accordingly. In conventional design methods for designing a metasurface and, thus, its units cells the level of approximation involved in assigning local reflection coefficients and/or transmission coefficients (depending on whether the metasurface is a reflection and/or transmission metasurface) to individual unit cells in the metasurface is unclear. Such conventional design methods may be only valid for slowly varying structures and may lead to inaccurate estimations of efficiencies when the meta-atom geometry of the metasurface changes rapidly from a unit cell to the next one, for instance in aperiodic or quasi-periodic structures of the metasurface. In addition, conventional approaches for designing metasurfaces merely focus on phase masking or phase gradient between two adjacent unit cells (i.e. the effect on the phase of electromagnetic waves by two adjacent unit cells or difference in the effect on the phase between the two adjacent unit cells) without any considerations for variations in their amplitudes (i.e. without considering the effect on the amplitude of electromagnetic waves by the two adjacent unit cells). The passage “directly next to each other” and the term “adjacent” may be sued as synonyms.

In other words, conventional design methods for designing a metasurface calculate the efficiency and the accuracy of reflection and/or transmission of electromagnetic waves by the metasurface to be designed based on the phase response of the unit cells in the metasurface to be designed. While the amplitude response of each unit cell of the metasurface may easily cause errors in the final reflection beam and/or transmission beam (when electromagnetic waves imping on the metasurface) as well as its efficiency. In conventional design methods, the amplitude response of each unit cell of the metasurface is expected or assumed to be a maximum possible value, which may be considered a value close to one, regardless of its effects in the beamforming process of electromagnetic waves impinging on the metasurface. Practically, this way of conventional designing produces a metasurface with a set of unit cells that have different reflection amplitudes and transmission amplitudes independent of each other. This may be resembled as an array of radiators whose elements radiate with different input powers. Due to this, in conventional design methods, there may be an error in the main beam of reflection and main beam of transmission (when electromagnetic waves imping on the metasurface) as well as an error in estimation of efficiency.

Furthermore, in conventional design methods, the phase response of each unit cell of a metasurface to be designed may not be as exact as the designer desires due to inconsistencies and limitations in the design procedure. In other words, the aforementioned error may be seen in phase response of each unit cell due to three reasons. First, it may come from an inconsistency between what the design desires and what is achievable numerically. For example, for achieving a unit cell with a phase response of 90-degree (i.e. 7t/2), the design of the unit cell should give or yield exactly this phase response. However, most of the time this is impossible because of the limitations in the physical attributes of the meta-atoms and unit cells of the metasurface. For the aforementioned desired phase response, as an example, usually a phase response between 85-degree to 95 degree may be considered as a proper response in the conventional design methods. However, these deviations in the phase response of unit cells may lead to a considerable error in the beamforming process.

Another reason for the aforementioned error with regard to the desired phase response of the unit cells in conventional design methods may lay in the realized phase response of each unit cell related to the numerical calculation. The realized phase response may be affected by fabrication process which changes the whole results. A further reason for the aforementioned error is that in conventional design methods the analysis of a metasurface is based on the Floquet-Bloch (FB) theory. To calculate the phase response of each unit cell in a periodic structure of a metasurface, the FB theory assumes that a unit cell is repeated periodically to form a periodic structure with infinite number of same unit cells. In this way, the mutual coupling between the unit cells, for instance for quasi-periodic structures, may not be a perfect estimation. Based on all the inconsistencies mentioned above, conventional design techniques are not sufficiently efficient for practical purposes. Based on the mentioned issues in the conventional design approach of metasurfaces, there are three main deficiencies in the performance of them. Usually a high specular reflection happens in metasurfaces when electromagnetic waves imping on the respective metasurface. This means a considerable part of the incident power (of the electromagnetic waves impinging on the metasurface) reflects back to the specular direction. In addition, due to the aforementioned different inconsistencies in the design process of conventional design methods, there may be different grating lobes in the reflection pattern as well as higher side lobe levels than the desired amount. Finally, mirror reflection may be present in metasurfaces that are conventionally designed.

In view of the above, this disclosure aims to provide an improved metasurface and an improved method for designing a metasurface. An objective of this disclosure is to provide a metasurface that overcomes at least one of the aforementioned deficiencies of metasurface designed using conventional design methods. Accordingly, it may be an objective to provide a method for designing a metasurface that overcomes at least one of the aforementioned deficiencies of conventional design methods.

These and other objectives are achieved by the solution of this disclosure as described in the independent claims. Advantageous implementations are further defined in the dependent claims.

A first aspect of this disclosure provides a metasurface for electromagnetic waves. The metasurface comprises unit cells; and each unit cell of the unit cells has a unique phase response and comprises multiple independent scatterers.

In other words, the unit cells (i.e. a phase gradient of the unit cells) obey Snell’s law. That is, the independent scatterers of each unit cell have the same phase response or collectively contribute to the unique phase response of the unit cell.

For instance, each unit cell of the unit cells of a super cell of the metasurface (i.e. of a group of unit cells covering a 2K phase shift) may have a unique phase response and comprise multiple independent scatterers. The unit cells of the super cell obey Snell’s law. That is, the phase gradient of the unit cells of the super cell obey Snell’s law. Optionally, the region of the metasurface, in which an independent scatterer is arranged or placed, may be referred to as “meta-atom”. The aforementioned region, i.e. the meta-atom, may be a periodic region. For instance, a meta-atom of the metasurface may be the smallest periodic region in the metasurface comprising one or more independent scatterers.

The metasurface may be considered as a homogenous medium for an incident electromagnetic wave impinging on the metasurface. In this case, the following condition is met:

In the above formula, U _c denotes the size of a unit cell and and m are lossless refractive indices of two half spaces on both sides of the metasurface. The metasurface may be a subwavelengthly thin periodic structure. The angle “0 ” is the incident angle to the normal vector of the metasurface plane. The term “m,2” in the above formula stands for for reflected diffraction orders of the incident electromagnetic wave and for m for transmitted diffraction orders of the incident electromagnetic wave.

A group of unit cells of the unit cells of the metasurface may cover a 2K phase shift, the unit cells of the group being arranged directly next to each other. In other words, a group of unit cells of the unit cells may cover a 2K phase shift in their responses, the unit cells of the group being arranged directly next to each other. The group of unit cells may be referred to as “super cell”. In other words, the unit cells in a super cell of the metasurface may cover a 2K phase shift. Within the aforementioned group, i.e. within the super cell, the unit cells of the group (i.e. super cell) have a different phase response with respect to each other. That is, each unit cell of the group has a different phase response with regard to the one or more other unit cells of the group. The passages “unit cells of a super cell” and “unit cells in a super cell” may be used as synonyms.

At least two unit cells of the unit cells of the metasurface may have a different phase response. This may be achieved in that the aforementioned at least two unit cells have a different number of independent scatterers and/or different forms of scatterers. In addition or alternatively, the different phase responses may be caused by any difference in the structure of the aforementioned unit cells that may cause a phase shift. For instance, at least two unit cells of a super cell (i.e. of a group of unit cells covering a 2K phase shift) may have a different phase response. This may be achieved in that the at least two unit cells of the super cell have a different number of independent scatterers and/or different forms of scatterers. In addition or alternatively, the different phase responses may be caused by any difference in the structure of the at least two unit cells of the super cell that may cause a phase shift.

In an implementation form of the first aspect, the multiple independent scatterers are arranged in a two-dimensional lattice.

The multiple independent scatterers arranged in the two-dimensional lattice allow forming a more homogeneous metasurface for the incident electromagnetic wave. The multiple independent scatterers arranged in the two-dimensional lattice allow a spatial oversampling for designing the metasurface. This improves designing the metasurface and, thus, the metasurface. The spatial oversampling allows to compensate errors caused during the design process of the metasurface and, thus, improves the metasurface that is generated by the design process.

In an implementation form of the first aspect, each unit cell of the unit cells is divided into subunit cells each comprising one or more independent scatterers of the multiple independent scatterers, wherein the phase responses of the sub-unit cells of the unit cell converge to the phase response of the unit cell.

For instance, each unit cell of a super cell of the metasurface may be divided into sub-unit cells each comprising one or more independent scatterers of the multiple independent scatterers, wherein the phase responses of the sub-unit cells of the unit cell converge to the phase response of the unit cell.

In the spatial oversampling for designing the metasurface (that is allowed due to the plurality of independent scatterers of the unit cell), each sub-unit cell may be interpreted as a quantizer by which the received signal (corresponding to the incident electromagnetic wave(s)) is to be quantized to a desired reflected or transmitted signal (electromagnetic wave(s)) in accordance to its transfer function. Thus, dividing the unit cells into sub-unit cells allows improving the metasurfaces because it supports the spatial oversampling for designing the metasurface and, thus, allows compensating errors caused by the design process of the metasurface. As a result, an improved metasurface may be achieved.

The unit cells of a super cell may comprise the same number of sub-unit cells. The number of scatterers in the sub-unit cells may be differently to each other. Optionally, a sub-unit cell may be a meta-atom of the metasurface. For instance, the sub-unit cells of a unit cell may be metaatoms of the metasurface. The sub-unit cells of a super cell may be meta-atoms.

In an implementation form of the first aspect, the multiple independent scatterers of each unit cell of the unit cells are arranged in a two-dimensional lattice. The unit cell may be divided into the sub-unit cells such that each sub-unit cell comprises a row and/or a column of one or more independent scatterers of the multiple independent scatterers.

For instance, the multiple independent scatterers of each unit cell of a super cell may be arranged in a two-dimensional lattice wherein the unit cell may be divided into the sub-unit cells such that each sub-unit cell comprises a row and/or a column of one or more independent scatterers of the multiple independent scatterers.

The multiple independent scatterers per unit cell of the metasurface allows more control on power of electromagnetic waves reflecting or transmitting from the metasurface. Arranging the multiple independent scatterers in a two-dimensional lattice allows more control on two dimensions, if applicable.

In an implementation form of the first aspect, each unit cell of the unit cells is divided into the sub-unit cells such that a number of the sub-unit cells equals a number of the multiple independent scatterers and each sub-unit cell of the sub-unit cells comprises a respective independent scatterer of the multiple independent scatterers.

In an implementation form of the first aspect, a group of unit cells of the unit cells cover a 2K phase shift, the unit cells of the group being arranged directly next to each other.

The passage “a group of unit cells of the unit cells cover a 2K phase shift” may be understood as meaning “a group of unit cells of the unit cells cover a 2K phase change in their responses”. The terms “phase shift” and “phase change” may be used as synonyms. The group of unit cells that cover a 2K phase shift may be referred to as super cell. In other words, the unit cells of a super cell may cover a 2K phase shift, wherein the unit cells of the super cell are arranged directly next to each other.

In an implementation form of the first aspect, the group of unit cells comprises 2 ^B unit cells of the unit cells, the 2 ^B unit cells of the group being arranged directly next to each other and B being an integer number greater than or equal to one.

In an implementation form of the first aspect, the group of the unit cells comprises a onedimensional arrangement of the sub-unit cells. The sub-unit cells of the group of unit cells may be designed such that a difference between a realized transfer response of a sub-unit cell and a desired transfer response of the sub-unit cell due to a design error when designing the sub-unit cell to achieve the desired transfer response is at least partly compensated by a difference between a desired transfer response of a second sub-unit cell, which is directly previous to the sub-unit cell, and a realized transfer response of the second sub-unit cell.

The aforementioned optional feature is an example of using spatial oversampling for designing the metasurface. Due to the aforementioned error compensation the metasurface is improved compared to a metasurface designed using a conventional design method, i.e. a design method without the concept of spatial oversampling. In the aforementioned case, a one-dimensional arrangement of the sub-unit cells of the group of the unit cells is considered for designing the metasurface.

In an implementation form of the first aspect, the difference between the realized transfer response of the sub-unit cell and the desired transfer response of the sub-unit cell due to the design error when designing the sub-unit cell to achieve the desired transfer response is at least partly compensated by adding, to the desired transfer response of the sub-unit cell, the difference between the desired transfer response of the second sub-unit cell and the realized transfer response of the second sub-unit cell.

In other words, the difference between the realized transfer response of the sub-unit cell and the desired transfer response of the sub-unit cell due to the design error when designing the subunit cell to achieve the desired transfer response is at least partly compensated by adding, to the desired transfer response of the sub-unit cell, the difference between the desired transfer response of the directly previous sub-unit cell and the realized transfer response of the directly previous sub-unit cell.

In an implementation form of the first aspect, the group of the unit cells comprises a two- dimensional arrangement of the sub-unit cells. The sub-unit cells of the group of unit cells may be designed such that a difference between a realized transfer response of a sub-unit cell and a desired transfer response of the sub-unit cell due to a design error when designing the sub-unit cell to achieve the desired transfer response is at least partly compensated by a difference between a desired transfer response of a second sub-unit cell, which is directly previous to the sub-unit cell in a first dimension, and a realized transfer response of the second sub-unit cell, and a difference between a desired transfer response of a third sub-unit cell, which is directly previous to the sub-unit cell in a second dimension, and a realized transfer response of the third sub-unit cell.

The aforementioned optional feature is an example of using spatial oversampling for designing the metasurface. Due to the aforementioned error compensation the metasurface is improved compared to a metasurface designed using a conventional design method, i.e. a design method without the concept of spatial oversampling. In the aforementioned case, a two-dimensional arrangement of the sub-unit cells of the group of the unit cells is considered for designing the metasurface.

In an implementation form of the first aspect, the difference between the realized transfer response of the sub-unit cell and the desired transfer response of the sub-unit cell due to the design error when designing the sub-unit to achieve the desired transfer response is at least partly compensated by adding, to the desired transfer response of the sub-unit cell, an average of the difference between the desired transfer response of the second sub-unit cell and the realized transfer response of the second sub-unit cell and of the difference between the desired transfer response of the third sub-unit cell and the realized transfer response of the third subunit cell.

Herein, the terms “first”, “second”, “third” etc. may be merely used for distinguishing elements from each other without necessarily providing an order to the elements. For example, the terms “second” and “third” of the aforementioned “second sub-unit cell” and “third sub-unit cell” are merely used to distinguish them from each other and other sub-unit cells. These terms are not used to specify a second or third sub-unit cell in e.g. a sequence of sub-unit cells.

In an implementation form of the first aspect, the realized transfer response of the sub-unit cell is designed to achieve the desired transfer response of the sub-unit cell using information on the design error and on a mutual coupling of the sub-unit cells of the group of unit cells.

Using information on the design error and on the mutual coupling of the sub-unit cells of the group of unit cells (i.e. a super cell) allows compensating errors caused by the design process and, thus, allows achieving an improved metasurface compared to a metasurface designed using a conventional design method.

In an implementation form of the first aspect, a respective transfer response of a respective subunit cell comprises an amplitude response and a phase response of the respective sub-unit cell.

Considering the amplitude response in addition to the phase response allows achieving an improved metasurface compared to a metasurface designed using a conventional design method.

In an implementation form of the first aspect, a desired transfer response of a respective subunit cell comprises an amplitude response of one and a phase response equalling to 2K divided by a number of the unit cells covering the 2K phase shift.

In an implementation form of the first aspect, the design error comprises an error due to an inconsistency of the design and an error due to assuming for the design a two-dimensional periodic structure of the unit cells of the group of unit cells having the same phase response.

For instance, the design error may comprise an error due to an inconsistency of the design and an error due to assuming for the design a two-dimensional periodic structure of the unit cells having the same phase response. The assumption may be that all the unit cells are considered as same to the designed unit cell which can cause the error. In an implementation form of the first aspect, the metasurface comprises multiple of the group of unit cells.

In other words, the metasurface comprises multiple of the super cell, i.e. multiple super cells. That is, when designing the metasurface multiple of the group of unit cells (i.e. multiple super cells) may be considered and, thus, the group of unit cells (i.e. the super cell) may be repeated in the metasurface. The number of the group of unit cells (i.e. the number of super cells) in the metasurface considered for designing the metasurface may be referred to as ultra-super cell size. That is, in case the metasurface has an ultra-super cell size of one, then merely the group (i.e. one group) of unit cells is considered; in case the metasurface has an ultra-super cell size of two, then two of the group of unit cells are considered and so on.

Increasing the number of the group of unit cells (i.e. number of the super cell), which is considered for designing the metasurface, above one (i.e. to two or more) allows increasing power in a desired reflection angle of electromagnetic waves impinging on the metasurface, and allows a reduction in the specular direction as well as a reduction of mirror reflection angle.

In order to achieve the metasurface according to the first aspect of this disclosure, some or all of the implementation forms and optional features of the first aspect, as described above, may be combined with each other.

A second aspect of this disclosure provides an antenna system comprising a metasurface according to the first aspect of this disclosure as described above.

The antenna system may comprise one or more antennas. The one or more antennas may be configured to transmit electromagnetic waves to the metasurface and/or receive electromagnetic waves from the metasurface. In other words, the metasurface may be configured to modulate electromagnetic waves transmitted by the one or more antennas and/or electromagnetic waves to be received by the one or more antennas.

The metasurface may be configured for beam steering of electromagnetic waves impinging on the metasurface. The one or more antennas and the metasurface may be arranged such that electromagnetic waves transmitted by the one or more antennas imping on the metasurface. In addition or alternatively, the one or more antennas and the metasurface may be arranged such that electromagnetic waves intended (e.g. transmitted) to the one or more antennas imping on the metasurface. In other words, the one or more antennas may be configured to transmit and/or receive electromagnetic waves via the metasurface. That is, the metasurface may be configured to modulate electromagnetic waves transmitted by the one or more antennas and/or electromagnetic waves received by the one or more antennas.

The description of the antenna system of the second aspect, e.g. of the metasurface, may be correspondingly valid for the metasurface according to the first aspect of this disclosure.

The antenna system of the second aspect and its implementation forms and optional features achieve the same advantages as the metasurface of the first aspect and its respective implementation forms and respective optional features.

In order to achieve the antenna system according to the second aspect of this disclosure, some or all of the implementation forms and optional features of the second aspect, as described above, may be combined with each other.

A third aspect of this disclosure provides a method for designing a metasurface for electromagnetic waves, wherein the method comprises providing in the metasurface unit cells, wherein each unit cell of the unit cells has a unique phase response and comprises multiple independent scatterers.

The description of the metasurface of the first aspect is correspondingly valid for the method of the third aspect.

In an implementation form of the third aspect, the method comprises arranging the multiple independent scatterers in a two-dimensional lattice.

In an implementation form of the third aspect, the method comprises dividing each unit cell of the unit cells into sub-unit cells each comprising one or more independent scatterers of the multiple independent scatterers, wherein the phase responses of the sub-unit cells of the unit cell converge to the phase response of the unit cell. In an implementation form of the third aspect, the method comprises arranging the multiple independent scatterers of each unit cell of the unit cells in a two-dimensional lattice, and dividing the unit cell into the sub-unit cells such that each sub-unit cell comprises a row and/or a column of one or more independent scatterers of the multiple independent scatterers.

In an implementation form of the third aspect, the method comprises dividing each unit cell of the unit cells into the sub-unit cells such that a number of the sub-unit cells equals a number of the multiple independent scatterers and each sub-unit cell of the sub-unit cells comprises a respective independent scatterer of the multiple independent scatterers.

In an implementation form of the third aspect, the method comprises providing a group of unit cells of the unit cells, wherein the unit cells of the group of unit cells cover a 2TI phase shift and are arranged directly next to each other.

In an implementation form of the third aspect, the method comprises providing the group of unit cells comprising 2 ^B unit cells of the unit cells, the 2 ^B unit cells of the group being arranged directly next to each other and B being an integer number greater than or equal to one.

In an implementation form of the third aspect, the group of the unit cells comprises a onedimensional arrangement of the sub-unit cells. The method may comprise designing a sub-unit cell of the group of unit cells to achieve a desired transfer response such that a difference between a realized transfer response of the sub-unit cell and the desired transfer response of the sub-unit cell due to a design error is at least partly compensated by a difference between a desired transfer response of a second sub-unit cell, which is directly previous to the sub-unit cell, and a realized transfer response of the second sub-unit cell.

In an implementation form of the third aspect, the method comprises at least partly compensating the difference between the realized transfer response of the sub-unit cell and the desired transfer response of the sub-unit cell due to the design error by adding, to the desired transfer response of the sub-unit cell, the difference between the desired transfer response of the second sub-unit cell and the realized transfer response of the second sub-unit cell.

In an implementation form of the third aspect, the group of the unit cells comprises a two- dimensional arrangement of the sub-unit cells. The method may comprise designing a sub-unit cell of the group of unit cells to achieve a desired transfer response such that a difference between a realized transfer response of the sub-unit cell and the desired transfer response of the sub-unit cell due to a design error is at least partly compensated by a difference between a desired transfer response of a second sub-unit cell, which is directly previous to the sub-unit cell in a first dimension, and a realized transfer response of the second sub-unit cell, and a difference between a desired transfer response of a third sub-unit cell, which is directly previous to the sub-unit cell in a second dimension, and a realized transfer response of the third sub-unit cell.

It may be assumed that the first dimension means same row and second dimension means same column. The differences may be applied for previous sub-unit cells in same row and same column.

In an implementation form of the third aspect, the method comprises at least partly compensating the difference between the realized transfer response of the sub-unit cell and the desired transfer response of the sub-unit cell due to the design error by adding, to the desired transfer response of the sub-unit cell, an average of the difference between the desired transfer response of the second sub-unit cell and the realized transfer response of the second sub-unit cell and of the difference between the desired transfer response of the third sub-unit cell and the realized transfer response of the third sub-unit cell.

In an implementation form of the third aspect, the method comprises designing the realized transfer response of the sub-unit cell to achieve the desired transfer response of the sub-unit cell using information on the design error and on a mutual coupling of the sub-unit cells of the group of unit cells.

In an implementation form of the third aspect, a respective transfer response of a respective sub-unit cell comprises an amplitude response and a phase response of the respective sub-unit cell.

In an implementation form of the third aspect, a desired transfer response of a respective subunit cell comprises an amplitude response of one and a phase response equalling to 2K divided by a number of the unit cells covering the 2K phase shift. In an implementation form of the third aspect, the design error comprises an error due to an inconsistency of the design and an error due to assuming for the design a two-dimensional periodic structure of the unit cells of the group of unit cells having the same phase response.

In an implementation form of the third aspect, the method comprises designing the metasurface by considering multiple of the group of unit cells.

That is, the method may comprise designing the metasurface by considering multiple super cells.

The method of the third aspect and its implementation forms and optional features achieve the same advantages as the metasurface of the first aspect and its respective implementation forms and respective optional features.

In order to achieve the method according to the third aspect of this disclosure, some or all of the implementation forms and optional features of the third aspect, as described above, may be combined with each other.

A fourth aspect of this disclosure provides a computer program comprising instructions which, when the program is executed by a computer, cause the computer to carry out the method according to the third aspect of this disclosure or any of its implementation forms.

In other words, a computer program is provided that comprises program code for performing when implemented on a processor, a method according to the third aspect of this disclosure or any of its implementation forms.

A fifth aspect of this disclosure provides a computer-readable storage medium comprising instructions which, when executed by a computer, cause the computer to carry out the method according to the third aspect of this disclosure or any of its implementation forms.

In other words, a computer readable storage medium is provided that stores executable program code which, when executed by a processor, causes the method according to the third aspect or any of its implementation forms to be performed. An sixth aspect of this disclosure provides a computer comprising a memory and a processor, which are configured to store and execute program code to perform the method according to the third aspect or any of its implementation forms.

A seventh aspect of this disclosure provides a non-transitory storage medium storing executable program code which, when executed by a processor, causes the method according to the third aspect or any of its implementation forms to be performed.

The computer program according to the fourth aspect, the computer-readable storage medium according to the fifth aspect, the computer according to the sixth aspect and the non-transitory storage medium according to the seventh aspect each achieve the same advantages as the metasurface of the first aspect and its respective implementation forms and respective optional features.

It has to be noted that all devices, elements, units and means described in the present application could be implemented in software or hardware elements or any kind of combination thereof. All steps which are performed by the various entities described in the present application as well as the functionalities described to be performed by the various entities are intended to mean that the respective entity is adapted to or configured to perform the respective steps and functionalities. Even if, in the following description of specific embodiments, a specific functionality or step to be performed by external entities is not reflected in the description of a specific detailed element of that entity which performs that specific step or functionality, it should be clear for a skilled person that these methods and functionalities can be implemented in respective software or hardware elements, or any kind of combination thereof.

BRIEF DESCRIPTION OF DRAWINGS

The above described aspects and implementation forms will be explained in the following description of specific embodiments in relation to the enclosed drawings, in which

Figure 1 shows an example of a metasurface according to an embodiment of this disclosure;

Figure 2 shows an example of a method according to an embodiment of this disclosure for designing a metasurface; Figure 3 shows an example of a metasurface according to an embodiment of this disclosure;

Figure 4 shows an example of sub-unit cells of a group of unit cells covering a 2K phase shift of a metasurface according to an embodiment of this disclosure, when assuming beam steering in one dimension;

Figure 5 shows an example of a method according to an embodiment of this disclosure for designing a metasurface, when assuming beam steering in one dimension;

Figure 6 shows an example of sub-unit cells of a group of unit cells covering a 2K phase shift (i.e. covering a 2K phase change in their phase response) in a metasurface according to an embodiment of this disclosure, when assuming beam steering in two dimensions;

Figure 7 shows an example of a method according to an embodiment of this disclosure for designing a metasurface, when assuming beam steering in two dimensions;

Figures 8 and 9 each show the normalized power pattern over the spatial angular domain for four different examples of a metasurface according to an embodiment of this disclosure; and

Figures 10 and 11 each show an example of a metasurface according to an embodiment of this disclosure.

In the Figures, corresponding elements are labeled with the same reference sign.

DETAILED DESCRIPTION OF EMBODIMENTS

Figure 1 shows an example of a metasurface according to an embodiment of this disclosure. The metasurface of Figure 1 is an example of the metasurface according to the first aspect of this disclosure. The description of the metasurface according to the first aspect is correspondingly valid for the metasurface of Figure 1.

The metasurface 1 of Figure 1 is a metasurface for electromagnetic waves. The metasurface 1 comprises unit cells 2 (in Figure 1 only one unit cell of such unit cells 2 is shown). Each unit cell 2 of the unit cells 2 has a unique phase response and comprises multiple independent scatterers 3. As shown in Figure 1, each unit cell 2 comprises k independent scatterers 3i to 3k, wherein k is an integer greater than or equal to two (k > 2). The multiple independent scatterers 3 of each unit cell 2 allow a spatial oversampling for designing the metasurface 1. This improves designing the metasurface 1 and, thus, the metasurface 1. The spatial oversampling allows to compensate errors caused during the design process of the metasurface 1 and, thus, improves the metasurface 1 that is generated by the design process. Further details with regard to the metasurface 1 are described with regard to the other Figures, for instance Figures 3, 10 and 11.

The metasurface 1 of Figure 1 may be part of an antenna system. For further details on such an antenna system reference is made to the description of the antenna system according to the second aspect of this disclosure.

Figure 2 shows an example of a method according to an embodiment of this disclosure for designing a metasurface. The method of Figure 2 is an example of the method according to the third aspect of this disclosure. The description of the method according to the third aspect is correspondingly valid for the method of Figure 2.

As shown in Figure 2, the method for designing a metasurface comprises providing in the metasurface unit cells, wherein each unit cell of the unit cells has a unique phase response and comprises multiple independent scatterers. For instance, in a group of unit cells of the unit cells of the metasurface the unit cells may cover a 2K phase shift, wherein each unit cell of the group of unit cells may have a unique phase response. The aforementioned group of unit cells (covering a 2K phase shift) may form a super cell of the metasurface, i.e. it may be referred to as “super cell”. In the metasurface the aforementioned group of unit cells, i.e. the super cell, may be repeated. Thus, in two or more super cells of the metasurface there may be corresponding unit cells that have a same unique phase response. The repetition of the super cell, i.e. of the group of unit cells covering a 2K phase shift, may be periodic. This is shown in Figure 2 as step SI of the method. Thus, the method of Figure 2 may be used for designing the metasurface 1 of Figure 1. Further details with regard to the method for designing a metasurface are described with regard to the other Figures, for instance Figures 4, 5, 6 and 7.

Figure 3 shows an example of a metasurface according to an embodiment of this disclosure. The metasurface 1 of Figure 3 is an example of the metasurface 1 of Figure 1. Thus, the description of Figure 1 is valid for the metasurface 1 of Figure 3 and in the following mainly additional optional features are described.

Figure 3 shows multiple unit cells 2 of the metasurface 1. According to Figure 3, three unit cells 2i, 2 ₂ and 2 _n are shown in detail. The number of unit cells is only by way of example and does not limit the present disclosure. Each unit cell 2 of the unit cells 2 of the metasurface 1 comprises multiple independent scatterers 3. The number of multiple independent scatterers 3 per unit cell 2 shown in Figure 3 is only by way of example and does not limit the present disclosure. The phase response of the unit cells 2 of the metasurface 1 obey Snell’s law. That is, the independent scatterers 3 of each unit cell 2 have the same phase response or collectively contribute to the unique phase response of the unit cell 2.

As shown in Figure 3, the multiple independent scatterers 3 of a respective unit cell 2 may be arranged in a two-dimensional lattice. There may be m columns of a independent scatterers 3 or a rows of m independent scatterers 3, wherein m and a are each an integer that is greater than or equal to two (m > 2, a > 2). In other words, m is the number of columns of independent scatterers 3 in the unit cell 2 and a is the number of rows of independent scatterers 3 in the unit cell 2.

The multiple scatterers 3 of a unit cell 2 of the metasurface 1 are independent. This is indicated in Figure 3 by the respective surface currents i _na m that are independent to each other. Optionally, at least two unit cells of the unit cells 2 of the metasurface 1 may have a different number of independent scatterers 3.

The following is true for the metasurface 1 of Figure 1 as well as the metasurface 1 of Figure 3. Each unit cell of the unit cells 2 may be divided into sub-unit cells each comprising one or more independent scatterers of the multiple independent scatterers 3, wherein the phase responses of the sub-unit cells of the unit cell converge to the phase response of the unit cell (not shown in Figures 1 and 3). In other words, the multiple independent scatterers 3 per unit cell 2 of the metasurface allow dividing each unit cell of the unit cells 2 of the metasurface 1 into sub-unit cells each comprising at least one independent scatterer of the multiple independent scatterers 3, wherein the phase responses of the sub-unit cells of the unit cell converge to the phase response of the unit cell. Optionally, each unit cell of the unit cells 2 may be divided into sub-unit cells which have the same phase response with the unit cell and comprise at least one scatterer of the multiple independent scatterers 3 of the unit cell.

For designing the unit cells 2 of the metasurface 1 the following three main assumptions based on different metasurface models and criteria may be satisfied. For instance, the phase gradient (phase change) among unit cells 2 of the metasurface 1 obey Snell’s law, which stipulates that each unit cell of the unit cells 2 has a unique phase response. This criterion leads to the case that the independent scatterers 3 of a unit cell 2 have the same phase response or collectively contribute to a unique phase response.

When designing the metasurface 1, the metasurface 1 may be considered as a homogenous medium for an incident electromagnetic wave impinging on the metasurface 1. In this case, the following condition is met:

In the above formula, U _c denotes the size of a unit cell and and m are lossless refractive indices of two half spaces on both sides of the metasurface. The metasurface 1 may be a subwavelengthly thin periodic structure. The angle “0 ” is the incident angle to the normal vector of the metasurface plane. The term “m,2” in the above formula stands for for reflected diffraction orders of the incident electromagnetic wave and for m for transmitted diffraction orders of the incident electromagnetic wave.

Thus, in a unit cell 2 comprising multiple independent scatterers 3 of the metasurface 1, there are more than one sample of received signal in the same aperture size of the unit cell 2 (when a signal is received in the form of electromagnetic waves impinging on the metasurface). This means that the received signal may be oversampled with a spatial sampling rate that is at least two times more compared to conventional metasurfaces (i.e. metasurfaces with unit cells each comprising only one independent scatterer). This oversampling depends on a dense ratio Dr of the unit cells 2 of the metasurface 1, wherein the dense ratio Dr of a unit cell of the unit cells 2 defines the number of sub-unit cells in the unit cell 2. That is, the greater the dense ratio Dr of a unit cell the greater the number of sub-unit cells and vice versa. Hence, each sub-unit cell of a unit cell 2 of the metasurface 1 may be interpreted as a quantizer by which the received signal should be quantized to the desired reflection signal and/or transmission signal in accordance to its transfer function.

Providing multiple independent scatterers 3 per unit cell 2 of the metasurface 1 may be referred to as dense unit cell concept. That is, due to the dense unit cell concept each unit cell of the unit cells 2 of the metasurface 1 comprise multiple independent scatterers 3. The dense unit cell concept (i.e. each unit cells comprising multiple independent scatterers 3) leads to two improvements: There may be a more accurate estimation of homogeneity of the metasurface 1. A further improvement of the dense unit cell concept lays in the possible spatial oversampling of the received signal, which may reduce the quantization error using faster-than-Nyquist spatial sampling. A unit cell comprising multiple independent scatterers may be referred to as “dense unit cell”.

Optionally, for designing the metasurface 1 with the unit cells 2 each comprising multiple independent scatterers, a Delta-Sigma concept may be exploited from a temporal signal processing to oversample the metasurface spatially. In temporal domain signal processing, which may be used in analog-to-digital converters, oversampling may push the quantization error to be seen at higher frequencies related to the central frequency of samples. Thus, the interested bandwidth can be retrieved by filtering using a low-pass filter. In this way, the unwanted noise can be omitted. Using a similar concept in spatial domain for designing the metasurface 1 allows to form the beam in metasurfaces by Delta-Sigma oversampling which force the undesired noise to invisible region of radiation in k-space or spatial angular domain. The aforementioned use of the Delta-Sigma concept in spatial oversampling for designing the metasurface 1 allows countering mutual coupling between unit cells each comprising multiple independent scatterers, e.g. between sub-unit cells. The mutual coupling increases with increasing dense ratio Dr of a unit cell of the metasurface. The mutual coupling may limit the improvement of beam pattern accuracy as well as efficiency.

Moreover, based on the conventional design (modelling) of a metasurface using the Floquet- Bloch theorem (FB theorem), the design procedure may not see the intrinsic error as the FB theorem finds the unit cell response of the unit cells 2 of the metasurface 1 by assuming a periodic structure with same unit cell. However, this assumption may not be true due to different interaction of each unit cell with incident electromagnetic waves (when the electromagnetic waves impinge on the metasurface 1). This mismatch between the designed unit cells and real response of them may cause an error in the response of the metasurface 1 which usually affects on the performance and efficiency of the metasurface 1. Further, in conventional design methods for designing a metasurface, there is not a correct estimation of mutual response of unit cells, for instance in metasurfaces with aperiodic unit cells when unit cells change very sharply. This mutual response of different unit cells may add another error to the final design as it is ignored in the conventional approach or it is assumed to be a constant value based on the FB theorem. The aforementioned errors may be countered by the aforementioned use of the Delta-Sigma concept in spatial oversampling for designing the metasurface. This is described in detail with regard to Figures 4 to 7, wherein in the examples of Figures 4 and 5 it is assumed that only one dimension of sub-unit cells of the metasurface is used for designing the metasurface, and in the examples of Figures 6 and 7 it is assumed that two dimensions of subunit cells of the metasurface are used for designing the metasurface.

In the following, with regard to an example of a method for designing a metasurface, steps to design a reflection metasurface are described. This description is correspondingly valid for designing a transmission metasurface. For the following description reference is made to the metasurface of Figure 1 or 3. Further, in this disclosure a unit cell comprising multiple independent scatterers may be referred to as “dense unit cell”. A group of unit cells of the dense unit cells 2 (each comprising multiple independent scatterers 3) of the metasurface 1 may cover a 2K phase shift, wherein the unit cells of the group are arranged directly next to each other. The passage “set of unit cells” may be used as a synonym for the passage “group of unit cells”. The group of unit cells covering a 2K phase shift may be referred to as “super cell”. The number of dense unit cells in a super cell may be computed by the number of desired bits (B) in the metasurface design. B may be an integer greater than or equal to one. In a super cell, there may be 2 ^B dense unit cells. Each dense unit cell may be divided into sub-unit cells each comprising one or more independent scatterers of the multiple independent scatterers of the dense unit cell. That is, the sub-unit cells may divide a dense unit cell to a group of m smaller sections, wherein m is an integer greater than or equal to two (m > 2). The sub-unit cells of the dense unit cell may have the same phase response ideally. Therefore, each super cell of the metasurface 1 may comprise 2 ^B ■ m sub-unit cells. The arrangement of the sub-unit cells may depend on the design requirements. In other words, the arrangement and topology of the sub-unit cells may be related to the beam steering dimensions, which may be in one or two dimensions, and a topology of the multiple independent scatterers in a dense unit cell. Assuming a rectangular two-dimensional lattice of multiple independent scatterers inside a unit cell, as exemplarily shown in Figure 3, there may be a number “a” of multiple independent scatterers in one sub-unit cell, wherein the unit cell 2 may comprise m sub-unit cells. In other words, the unit cell 2 may comprise m sub-unit cells, wherein each sub-unit cell may comprise a independent scatterers, wherein a and m are each integers that are greater than or equal to two (a > 2, m > 2). Thus, there may be a ■ m ■ 2 ^B scatterers in a super cell of a metasurface according to Figure 1 or 3, whereas in a conventional metasurface comprising only one independent scatterer per unit cell there may be only 2 ^B scatterers in a super cell. Based on this, the dense ratio Dr of a unit cell comprising multiple independent scatterers (i.e. of a dense unit cell) may be defined as the number of sub-unit cells in the unit cell (i.e. dense unit cell). For achieving a proper reflection or refraction in a metasurface, the unit cells forming a super cell are covering a 2K phase shift (2K phase change) and the phase gradient in a super cell obey the generalized Snell's law (GSL) as shown in the following:

2 71 o x

In the above formula the terms “n” and “n _r “ are the refractive indices of the input and output media, respectively and the terms “0 ” and “9 _r ” are the propagation angles of the incident and refracted electromagnetic waves, respectively, when electromagnetic waves imping on the metasurface. For example, for the reflection mode of operation in free space (i.e. on both sides of the metasurface there is free space), the terms “n” and “n _r “ are the refractive indices of air (ni = n _r = 1).

Assuming an one-dimensional beam steering, as exemplarily shown in Figure 4, only phase change in the plane that is parallel with the beam steering plane may be valuable and therefore, sub-unit cells in a unit cell may be chosen as columnar arrays of independent scatterers perpendicular to that plane. In this way, the number of columnar arrays in a unit cell may determine the dense ratio of the design. Following the generalized Snell's law, it may be assumed that each of the aforementioned sub-unit cells have a same response. Optionally, this means that each of the aforementioned sub-unit cells ideally have an amplitude response close to or equal to unity and a phase response according to the generalized Snell's law. This ideal response is shown by the term “g _n in the following formulas: Ignl = 1, while t = 0, 1, (2 ^B - 1)

In the above formulas, the term “g _n ” is an ideal transfer response of a unit cell. This is not its realized response due to some errors existing in the design process of the metasurface. |g _n | and g _n are ideal (desired) amplitude and phase of each sub-unit cell respectively. The term cpo denotes initial phase in a first sub-unit cell of sub-unit cells of a unit cell. The passage “2 ^B ” denotes the number of unit cells in a super cell, i.e. the number of unit cells in the metasurface that cover a 2K phase shift.

As already mentioned, there may be some errors in the design process. For instance, the designed unit cell may not be perfectly matched to the requirements and there may be a slight inconsistency in amplitude response and phase response of the ideal transfer response of a unit cell and the designed (desired) unit cell. A numerical analysis used in the design process may consider the incident electromagnetic wave (when electromagnetic waves imping on the meta surface) as perfectly plane waves. This means there is no estimation for waves with arbitrary wave front. The mutual response of the sub-unit cells in a super cell of the metasurface may meet an increase using the dense unit cell concept.

To address these issues, this disclosure proposes a design method which may control the errors and allows designing the unit cells in accordance to these main criteria including their amplitude response, phase response, their mutual response and finally the whole performance of the metasurface not only based on the phase response of them. The following proposed technique may design a metasurface based on the mentioned criteria and links them together to optimize the outcome. For instance, the aforementioned Delta-Sigma concept helps to optimize the design method that may be based on the spatial oversampling of the unit cells, because the unit cells comprise multiple independent scatterers.

Figure 4 shows an example of sub-unit cells of a group of unit cells covering a 2K phase shift of a metasurface according to an embodiment of this disclosure, when assuming beam steering in one dimension.

In Figure 4, the sub-unit cells 4 of the unit cells of a metasurface forming a group of unit cells covering a 2K phase shift (i.e. the sub-unit cells 4 of a super cell) are shown, assuming that the design is done with regard to one dimension. In the following the group of unit cells covering a 2K phase shift is referred to as super cell. In Figure 4, the sub-unit cells 4i, 42 and 4N are exemplarily shown for the sub-unit cells of the unit cells 2 of a group of unit cells covering a 2K phase shift, wherein the unit cells of the group are arranged directly next to each other and, thus, the aforementioned sub-unit cells are arranged directly next to each other. The number of sub-unit cells is only by way of example and does not limit the present disclosure. There may be N = 2 ^B -Dr sub-unit cells in the group of unit cells covering a 2K phase shift, wherein 2 ^B is the number of unit cells of the group of unit cells covering a 2K phase shift (B being a integer greater or equal to one, B > 1) and Dr is the dense ratio of a unit cell of the unit cells of the metasurface that defines the number of sub-unit cells in the unit cell.

Figure 4 exemplarily shows an incident electromagnetic wave 5a when electromagnetic waves impinge on the metasurface and, thus, on the sub-unit cells 4 of the super cell exemplarily shown in Figure 4. Further, the corresponding reflected electromagnetic wave 5b reflected from the metasurface is shown. The angles “0 ” and “9 _r ” denote the propagation angles of the incident beam 5a and refracted beam 5b, respectively.

At the bottom of Figure 4 an example of the realized reflection coefficients Vi, V2, ..., VN of each sub-unit cell 4i, 42, ... , 4N of the super cell is shown. The value of the realized reflection coefficient may be between zero and one (0 < V _n < 1). The x-axis is the index n of the sub-unit cells 4 of the super cell, wherein n is greater than or equal to one and smaller than or equal to N = 2 ^B -Dr. The y-axis is the transfer response TR.

For achieving the example of realized reflection coefficients of the sub-unit cells 4 of the super cell shown in Figure 4, the method of Figure 5 for designing a metasurface may be performed.

Figure 5 shows an example of a method according to an embodiment of this disclosure for designing a metasurface, when assuming beam steering in one dimension. For the description of Figure 5 reference is made to the super cell of Figure 4, i.e. the group of unit cells of the unit cells of the metasurface that cover a 2K phase shift. In the Figures, such as Figure 5, the abbreviation “TR” stands for “transfer response”, the abbreviation “MR” stands for “mutual response” and the abbreviation “SUC” stands for “sub-unit cell”. The method of Figure 5 is based on a recursive design algorithm based on the Delta-Sigma concept. In this algorithm, the goal is to convert the desired transfer responses g _n of the subunit cells 4 of the super cell to an optimized series of complex transfer responses V _n (i.e. realized transfer responses), which are calculated to minimize the errors. Hence, in the algorithm diagram of Figure 5, g _n denotes the input series that are the ideal desired transfer responses of the sub-unit cells of the super cell. Moreover, the W _n series are the transfer responses of the sub-unit cells 4 in the design procedure (i.e. the design transfer responses) and the V _n series are the realized transfer responses of the sub-unit cells 4, which may take into account the errors due to inconsistency ei in the sub-unit cells 4 of the super cell and mutual response es of the sub-unit cells 4 of the super cell. In Figure 5, the S _n series are the outcome of numerical modelling of the design using the Floquet-Bloch theorem. As may be seen in the diagram, in each stage of the algorithm there is one Delta process and one Sigma process.

The Delta process subtracts the realized transfer response V _n of a sub-unit cell 4 _n of the subunit cells of the super cell from the desired transfer response g _n to compute an error in the n-th sub-unit cell 4 _n . This error is shown by e _n . The Sigma process adds this error e _n to the next desired transfer response g _n +i, which denotes the transfer response of the adjacent sub-unit cell 4 _n +i, i.e. the n+l-th sub-unit cell.

In other words, the method of Figure 5 for designing a metasurface assumes that the super cell, i.e. group of the unit cells covering a 2K phase shift, comprises a one-dimensional arrangement of the sub-unit cells. The method of Figure 5 comprises designing a sub-unit cell 4 _n of the group of unit cells (i.e. super cell) to achieve a desired transfer response g _n such that a difference between a realized transfer response V _n of the sub-unit cell 4 _n and the desired transfer response g _n of the sub-unit cell 4 _n due to a design error is at least partly compensated by a difference e _n - i between a desired transfer response g _n -i of a second sub-unit cell 4 _n -i, which is directly previous to the sub-unit cell 4 _n , and a realized transfer response V _n -i of the second sub-unit cell 4 _n -i. With regard thereto, the method of Figure 5 comprises at least partly compensating the difference between the realized transfer response V _n of the sub-unit cell 4 _n and the desired transfer response g _n of the sub-unit cell 4 _n due to the design error by adding, to the desired transfer response g _n of the sub-unit cell 4 _n , the difference between the desired transfer response gn-i of the second sub-unit cell 4 _n -i and the realized transfer response V _n -i of the second subunit cell 4 _n -i. As indicated in Figure 5, the realized transfer response V _n of the sub-unit cell 4 _n is designed to achieve the desired transfer response g _n of the sub-unit cell 4 _n using information on the design error and on a mutual coupling of the sub-unit cells of the group of unit cells (i.e. of the super cell). A desired transfer response of a respective sub-unit cell 4 _n may comprise an amplitude response of one and a phase response equalling to 2TI divided by the number of the unit cells (i.e. 2 ^B ) covering the 2TI phase shift, i.e. the number of unit cells of the super cell. The design error may comprise an error ei( _n ) due to an inconsistency of the design and an error es(n) due to assuming for the design a two-dimensional periodic structure of the unit cells of the group of unit cells having the same phase response. That is, the error es(n) is due to numerical modelling of the design using the Floquet-Bloch theorem.

Therefore, the algorithm used by the method of Figure 5 for designing a metasurface lets the design process to take the inconsistency, which comes from imperfect sub-unit cell design and modelling inefficiencies (due to using the Floquet-Bloch theorem), into account and compensates it by a negative feedback. The mutual responses originated from other adjacent sub-unit cells have been seen in the algorithm to be alleviated by the process. As mentioned above, the generalized Snell’s law (GSL) stipulates that all the sub-unit cells in a unit cell having multiple independent scatterers (i.e. a dense unit cells) may have the same transfer response and, all of them may have the same desired transfer response g _n . The sub-unit cells in a unit cell may collectively contribute to the unique phase response of the unit cell.

The Delta and Sigma processes of the algorithm of Figure 5 may be vector operations so that they may control amplitude and phase at the same time. As shown in Figure 5, the output W _n of the Sigma process for the n-th sub-unit cell 4 _n may be considered as updated desired transfer response of the n-th sub-unit cell 4 _n before considering the design error (e.g. error ei( _n ) due to an inconsistency of the design and an error es(n) due to assuming for the design a two- dimensional periodic structure of the unit cells of the group of unit cells having the same phase response). In this stage, based on the value of W _n , the designer for designing the metasurface is to come up with a physical structure which may produce the nearest possible transfer response to the W _n . This may be done according to a conventional method for designing a metasurface.

Next, the transfer response of the n-th sub-unit cell 4 _n may be calculated using a conventional design method based on the Floquet-Bloch theory to obtain S _n . For considering the correct mutual response, the effect of this error may be removed, which is done in the mutual response correction stage (which is indicated in Figure 5 for the n-th sub-unit cell 4 _n by the block “MR(n) correction”). Next, the algorithm of Figure 5 may calculate the realized response V _n of the n-th sub-unit cell 4 _n by adding the correct mutual response to S _n . This process is shown in the algorithm diagram of Figure 5 as the “Response correction”. For the n-th sub-unit cell 4 _n the final value of the transfer response (i.e. the realized transfer response) is V _n , which is the real behavior of the n-th sub-unit cell 4 _n by interaction with the incident electromagnetic wave. For the next stage (i.e. the n+l-th sub-unit cell 4 _n +i), the realized V _n of the n-th sub-unit cell 4 _n is subtracted from the desired transfer response g _n of the n-th sub-unit cell 4 _n and the result e _n (being an error) is added to the desired transfer response g _n +i of the next sub-unit cell, i.e. the n+l-th sub-unit cell 4 _n +i. The aforementioned steps with regard to the n-th sub-unit cell 4 _n steps may be continued at least for one super cell. Optionally, the algorithm may be continued for a greater number of super cells for better performance. To compare the effect of this, the beam pattern of the reflected wave may be modelled. Having calculated the beam pattern, the error between the designed and desired beam patterns determines a proper number of super cells for achieving the better performance. Further details with regard thereto are described with respect to Figure 11.

In a stage of the algorithm of the method of Figure 5 for designing a metasurface, it may be written that the realized transfer response V _n -i is as follows: V _n -i = gn-i + e _n -i. In a next stage, the realized transfer response V _n is as follows: V _n = gn + gn-i - V _n -i + e _n . By replacing V _n -i in the aforementioned formula, the realized transfer response V _n of the next stage may be written as: V _n = gn + gn-i - gn-i - e _n + Cn+i. As may be seen from the aforementioned formula, the errors, which are produced in the adjacent stages of the algorithm, are subtracted from each other so that they compensate each other. In other words, a difference e _n between the realized transfer response V _n of a sub-unit cell 4 _n (n-th sub-unit cell) and a desired transfer response g _n of the sub-unit cell 4 _n due to errors ei( _n ) and es(n) of the design procedure may be at least partly compensated by a difference e _n -i between a desired transfer response g _n -i of a second sub-unit cell (n-l-th sub-unit cell), which is directly previous to the sub-unit cell 4 _n , and a realized transfer response V _n -i of the second sub-unit cell.

As shown above, this may be achieved due to the Delta-Sigma concept, which allows shaping the errors without using more accurate sub-unit cells or increasing a number of unit cells (decreasing phase gradient of unit cells). The Delta-Sigma concept may be used because the unit cells 2 of the metasurface 1 comprise multiple independent scatterers, i.e. they are dense unit cells. In contrast thereto, conventional metasurfaces have unit cells each comprising only one independent scatterer and conventional methods for designing a metasurface do not use the Delta-Sigma concept. The above described algorithm for one-dimensional beam steering may also be adapted for two-dimensional beam steering. That is the Delta-Sigma concept for designing a metasurface may be applied for two-dimensional beam steering. This is described in the following with respect to Figures 6 and 7.

Figure 6 shows an example of sub-unit cells of a group of unit cells covering a 2K phase shift (i.e. covering a 2K phase change in their phase response) in a metasurface according to an embodiment of this disclosure, when assuming beam steering in two dimensions. Figure 6 corresponds to Figure 4, wherein according to Figure 6 a two-dimensional arrangement of the sub-unit cells of the group of the unit cells covering a 2K phase shift (i.e. the super cell) is considered for designing the metasurface. The description of Figure 4 is correspondingly valid for Figure 6 and in the following mainly additional optional feature or differences are described.

As it is shown in Figure 6, for a two-dimensional beam steering, the super cell may have a two- dimensional phase gradient (PG) according to the desired reflection angle. These phase gradients PG are exemplarily depicted in Figure 6 in x-direction and y-direction and may be calculated separately using the generalized Snell’s law (GSL). Based on the GSL formula, the number of sub-unit cells of the super cell may be obtained for x-direction and y-direction, separately. This allows forming a two-dimensional super cell. The transfer response sampling may be carried out as a two-dimensional matrix of weights as illustrated in Figure 6. The process to generate the response samples may be made up of three stages: Obtaining horizontal response samples (RS) in a first row of a two-dimensional super cell. Obtaining vertical response samples (RS) in a first column of the two-dimensional super cell. Calculating individual response samples of sub-unit cells of a super cell (assuming a two-dimensional arrangement of the sub-unit cells) based on the two-dimensional algorithm according to the Delta-Sigma concept, as shown in Figure 7. In Figure 6, the different sub-unit cells 4 of the super cell are numbered by the indices “q” and “p”, wherein “q” is the index for the y-axis and “p” is the index for the x-axis. Each of the indices “q” and “p” may be greater than or equal to 1 and smaller than or equal to 2 ^B -Dr, wherein (in each direction of the two-dimensional arrangement) 2 ^B is the number of unit cells of the group of unit cells covering a 2K phase shift (B being a integer greater or equal to one, B > 1) and Dr is the dense ratio of a unit cell of the unit cells of the metasurface that defines the number of sub-unit cells in the unit cell. At the right side of Figure 6 an example of the realized reflection coefficients Vn, V12, Vqp of each sub-unit cell 4n, 4n, . . 4QP of the super cell is shown. For achieving the example of realized reflection coefficients of the sub-unit cells 4 of the super cell shown in Figure 6, the method of Figure 7 for designing a metasurface may be performed.

Figure 7 shows an example of a method according to an embodiment of this disclosure for designing a metasurface, when assuming beam steering in two dimensions. Figure 7 corresponds to Figure 5, wherein according to Figure 7 a two-dimensional arrangement of the sub-unit cells of the group of the unit cells is considered for designing the metasurface. The description of Figure 5 is correspondingly valid for Figure 7 and in the following mainly additional optional feature or differences are described.

The algorithm of Figure 5 regarding one dimension may calculate the error due to designing the metasurface in one dimension, due to the one-dimensional nature of the Delta-Sigma concept used in the algorithm of Figure 5. Hence, for a two-dimensional matrix (or lattice) of sub-unit cells, as shown in Figure 6, the algorithm of Figure 5 may minimize the error in one direction. To apply the Delta-Sigma concept to a two-dimensional case, a two-dimensional algorithm is proposed in Figure 7 that similarly exploits the one-dimensional Delta-Sigma concept of Figure 5. The two-dimensional algorithm of Figure 7 starts generation of response samples by considering a sub-unit cell 4qp, wherein for the first row (same x-axis value) and first column (same y-axis value) the one-dimensional algorithm of Figure 5 may be used. As shown in Figures 6 and 7, for a sub-unit cell 4 _qp of the super cell, the two-dimensional algorithm may consider the previous adjacent neighbours in both directions, that is the previous adjacent neighbour sub-unit cell 4 _q -i, _P in the y-direction (vertical neighbour) and the previous adjacent neighbour sub-unit cell 4 _q , _p -i in the x-direction (horizontal neighbour). As shown in Figure 7, with regard to the previous adjacent neighbour sub-unit cell 4 _q -i, _P in the y-direction the Delta process subtracts the realized transfer response V( _q -i) _P of the previous adjacent neighbour subunit cell 4 _q -i, _P in the y-direction from the desired transfer response g( _q -i) _P of the previous adjacent neighbour sub-unit cell 4 _q -i, _P in the y-direction to compute an error in the previous adjacent neighbour sub-unit cell 4 _q -i, _P in the y-direction. This error is shown by e( _q -i) _P . As shown in Figure 7, with regard to the previous adjacent neighbour sub-unit cell 4 _q , _p -i in the x-direction the Delta process subtracts the realized transfer response V _q ( _P -i) of the previous adjacent neighbour subunit cell 4q, _p -i in the x-direction from the desired transfer response g _q ( _P -i) of the previous adjacent neighbour sub-unit cell 4 _q , _p -i in the x-direction to compute an error in the previous adjacent neighbour sub-unit cell 4 _q , _p .i in the x-direction. This error is shown by e _q ( _P -i). In other words, the two-dimensional algorithm may calculate in a Delta process for the previous neighbour subunit cell 4 _q -i, _P in the y-direction and previous neighbour sub-unit cell 4 _q , _p -i in the x-direction the errors e( _q -i) _P and e _q ( _P -i), respectively. These errors e( _q -i) _P and e _q ( _P -i) may be averaged to provide an average error for the previous neighbour sub-unit cells in both directions as a common Delta output. Thus, the common Delta output (i.e. averaged errors) comprises the information on the errors e( _q -i) _P and e _q ( _P -i) regarding the previous neighbour sub-unit cells in both directions. The Sigma process adds this averaged error, i.e. the common Delta output, error to a desired transfer response gqp, which denotes the desired transfer response of the sub-unit cell 4 _q , _p . From this stage, the rest of the process is as outlined in the algorithm of Figure 5 to produce the realized transfer response V _qp for the sub-unit cell 4 _q , _p . The aforementioned algorithm steps may be repeated for all sub-unit cells of the two-dimensional arrangement (i.e. 4 _q , _p with 1 < q,p < 2 ^B -Dr) to define all the response samples (RS), i.e. transfer responses, in one supercell for a desired reflection. All the samples may be vectors and, thus, contain amplitude and phase. This means that all the operations may be vector operations.

In other words, as exemplarily shown in Figure 7, for designing a metasurface, it may be assumed that a group of unit cells covering a 2K phase shift (i.e. a super cell) comprises a two- dimensional arrangement of the sub-unit cells. The method for designing the metasurface may comprise designing a sub-unit cell 4 _q , _p of the group of unit cells to achieve a desired transfer response gqp such that a difference between a realized transfer response V _qp of the sub-unit cell 4 _q , _p and the desired transfer response gqp of the sub-unit cell 4 _q , _p due to a design error is at least partly compensated by a difference e _q ( _P -i) between a desired transfer response g _q ( _P -i) of a second sub-unit cell 4 _q , _p -i, which is directly previous to the sub-unit cell 4 _q , _p in a first dimension (e.g. in the x-direction), and a realized transfer response V _q ( _P -i) of the second sub-unit cell 4 _q , _p -i, and a difference e( _q -i) _P between a desired transfer response g( _q -i) _P of a third sub-unit cell 4 _q . i, _p , which is directly previous to the sub-unit cell 4 _q , _p in a second dimension (e.g. in the y-direction), and a realized transfer response V( _q -i) _P of the third sub-unit cell 4 _q -i, _P .

It may be assumed that the first dimension means same row and second dimension means same column. The differences may be applied for previous sub-unit cells in same row and same column. As shown in Figure 7, the method comprises at least partly compensating the difference between the realized transfer response V _qp of the sub-unit cell 4 _q , _p and the desired transfer response gqp of the sub-unit cell 4 _q , _p due to the design error by adding, to the desired transfer response g _qp of the sub-unit cell 4 _q , _p , an average of the difference e _q ( _P -i) between the desired transfer response g _q ( _P -i) of the second sub-unit cell 4 _q , _p -i and the realized transfer response V _q ( _P . i) of the second sub-unit cell 4 _q , _p -i and of the difference e( _q -i) _P between the desired transfer response g( _q -i) _P of the third sub-unit cell 4 _q -i, _P and the realized transfer response V( _q -i) _P of the third sub-unit cell 4 _q -i, _P .

With regard to Figure 7, for analysing a performance of the two-dimensional algorithm the following relations are true for Wqp and V _qp , wherein “Wqp” denotes the transfer response of the sub-unit cell 4 _q , _p in the design procedure and “Vqp” denotes the realized transfer response of the sub-unit cell 4 _q , _p .

By substituting the following relations into the aforementioned formula regarding the realized transfer response V _qp of the sub-unit cell 4 _q , _p q(p-l) ⁼ gq(p-l) ^{+ e} q(p-l) V(q-i)p ⁼ §(q-i)p ^{+ e} (q-i)p the formula regarding the realized transfer response Vqp of the sub-unit cell 4 _q , _p may be simplified as

Using the formula for normalized array factor, the power pattern for a two-dimensional periodic structure of sub-unit cells of a super cell of a metasurface with discrete weights Vqp may be written as

In the above formula, the terms “\|/ _x ” and “ \|/ _y ” are as outlined in the following formulas

In the above formulas, the terms “ko”, “d _x “ and “d _y ” are free space wave number, size of a subunit cell in x-direction and size of a sub-unit cell in y-direction, respectively. The reflection parameters “0 _r ” and “cp _r ” are two-dimensional reflection angles in relation to z-axis (0 _r ) and x- axis (cpr), respectively, in the order as shown in Figure 6.

Inserting, the above simplified formula regarding the realized transfer response V _qp of the sub- unit cell 4 _q , _p in the above formula for normalized array factor, an actual radiation pattern may be retrieved as shown in the following:

In the above equation, it is possible to separate the ideal pattern and the error pattern which affect directly on the performance of the metasurface. Due to the intrinsic characteristic of the above described Delta-Sigma concept, the average of errors (i.e. - (e _q ( _p-1 ) + e^.- p)) in the previous neighbour sub-unit cells 4 _q , _p -i and 4 _q -i, _P of a respective sub-unit cell 4 _q , _p is subtracted from the error eqp in the respective sub-unit cell 4 _q , _p , which is adapted to the Delta-Sigma concept. The Delta-Sigma technique may shape the noise to the invisible region of radiation pattern. As mentioned already above with regard to the one dimensional Delta-Sigma algorithm, the outcome of the Delta-Sigma algorithm may be improved by considering more than one super cells, i.e. considering for the algorithm the sub-unit cells of two or more super cells. This provides more samples for the algorithm and, thus, allows converging towards the best response asymptotically. A two-dimensional ultra-super cell may be used for the designing of the metasurface, wherein the two-dimensional ultra-super cell may be a fundamental periodic block, which comprises two or more super cells (instead of one super cell).

As shown and described above with regard to Figures 4, 5, 6 and 7, the Delta-Sigma algorithm for beamforming may determine the coefficients of each sub-unit cell based on the Delta-Sigma concept to alleviate the effect of mutual coupling in a metasurface. The Delta-Sigma algorithm may control the phase response as well as amplitude response to compensate errors and inconsistency in phase and amplitude of each sub-unit cell. Thus, the method for designing a metasurface as exemplarily described above with regard to Figures 4, 5, 6 and 7 may push the noise and unwanted power to the invisible region of radiation which improve the efficiency of the metasurface significantly.

Figures 8 and 9 each show the normalized power pattern over the spatial angular domain for four different examples of a metasurface according to an embodiment of this disclosure.

To analyze the performance of a metasurface that may be generated by a method for designing a metasurface according to this disclosure, as exemplarily described with regard to Figures 4, 5, 6 and 7, three different scenarios are compared in Figure 8. That is, Figure 8 shows three normalized power patterns of electromagnetic waves (e.g. a beam) reflected from the metasurface for three different scenarios. Figure 8 (a) shows the normalized power pattern for an ideal scenario, in which the unit cells of the metasurface radiate perfectly, mutual coupling between the unit cells is neglected and the unit cells reflect an incident normal electromagnetic wave to a desired reflecting angle of 60 degrees. This is a scenario, in which each unit cell radiates exactly a required phase with only one independent scatterer (e.g. meta-atom) and has a reflection coefficient of one. This scenario is artificial (i.e. does not exist in real life). It depicts a reflecting metasurface with 100% efficiency and is merely an example of a benchmark for the comparison purpose. Figure 8 (b) shows a scenario, in which the power pattern of electromagnetic waves impinging on the metasurface analyzed with one scatterer in each unit cell, which means a dense ratio of one (Dr = 1), as it is the case in the scenario of Figure 8 (a). However, in the scenario of Figure 8 (b), mutual coupling between unit cells and, thus, scatterers are calculated and considered. Since, the distance between scatterers may be less than half wavelength based on the generalized Snell’s law, it is acceptable that the efficiency is reduced, as shown in Figure 8 (b). This is the case with passive reflection metasurfaces in real life. This is one reason that reflecting metasurfaces are low efficient practically. In the third scenario, shown in Figure 8 (c), the metasurface according to the scenario of Figure 8 (a) is investigated, with the difference that each unit cell of the metasurface comprises two independent scatterers (i.e. the metasurface comprises dense unit cells). In the third scenario the dense ratio is assumed to be two (Dr = 2), i.e. each unit cell has two sub-unit cells, wherein each sub-unit cell comprises a respective scatterer of the two independent scatterers of the unit cell, and the realized mutual coupling between the sub-unit cells and, thus, the independent scatterers is considered. Further, in the third scenario the discrete coefficients are assigned based on the Delta-Sigma technique, as exemplarily described with regard to Figures 4 to 7, to alleviate the mutual coupling. In other words, Figure 8 (c) shows an example of a metasurface with unit cells comprising multiple independent scatterers that is generated by an example of a method according to this disclosure for designing a metasurface.

All the power patterns of Figure 8 (a), (b) and (c) are normalized to the maximum power of the scenario of Figure 8 (c). The y-axis of the graphs of Figure 8 are the normalized power pattern in dB and the x-axis of the graphs are the spatial angular domain. The term “K” is wave number and its unit is radian/meter. It denotes the spatial frequency of an electromagnetic wave. The term “Ko” is the wave number or spatial frequency of an electromagnetic wave in free space and is usually considered as the reference value of wave number. The term “K _x ” is the wave number of a surface electromagnetic wave on the metasurface structure in x direction. For the comparison purpose, the wave number in different materials is optionally expressed in relation to the wave number in free space. As may be retrieved from Figure 8, the power pattern in the scenario of Figure 8 (c) has the highest value in the desired direction and the specular reflection is less than -20 dB compared to the main beam reflection. On the other hand, the mirror reflection is less -14 dB compared to the main reflection as well. In the scenario of Figure 8 (c) it is assumed that for the design of the metasurface only one super cell is considered, that is the design is based on a ultra-super cell size of one (USC size = 1), which means the Delta-Sigma algorithm is only implemented to one super cell. The ultra-super cell size indicates the number of super cells that are considered when designing a metasurface using the Delta-Sigma algorithm. As described previously, it is possible to obtain higher performance, in case the Delta-Sigma algorithm is applied to more than one super cell (i.e. the ultra-super cell size is greater than one, USC size > 1). In this way, the Delta-Sigma algorithm may determine the transfer response of all the sub-unit cells in an ultra-super cell. To illustrate the impact of considering more than one super-cell, i.e. considering an ultra-super cell with a size greater than one (USC size > 1), Figure 9 compares the normalized power patterns for four different USC sizes (USC size = 1, 2, 3 and 4. As shown in Figure 9, greater USC size brings about more power in the desired reflection angle and on the other hand causes significant reduction in the specular direction as well as mirror reflection angle. The y-axis of the graphs of Figure 9 are the normalized power pattern in dB and the x-axis of the graphs are the spatial angular domain. The term “K” is wave number and its unit is radian/meter. It denotes the spatial frequency of an electromagnetic wave. The term “Ko” is the wave number or spatial frequency of an electromagnetic wave in free space and is usually considered as the reference value of wave number. The term “K _x ” is the wave number of a surface electromagnetic wave on the metasurface structure in x direction. For the comparison purpose, the wave number in different materials is optionally expressed in relation to the wave number in free space.

Figures 10 and 11 each show an example of a metasurface according to an embodiment of this disclosure.

The metasurface of Figure 10 is an example of the metasurface of Figure 1 and, thus, the description of Figure 1 is correspondingly valid for the metasurface of Figure 10. Figure 10 shows on the left side an example of a unit cell 2 of the metasurface. As shown, the unit cell 2 comprises two sub-unit cells 4, wherein each sub-unit cell 4 comprises two independent scatterers 3. That is, the four independent scatterers 3 of the unit cell 2 may be divided among two sub-unit cells 4, such that each sub-unit cell 4 comprises two independent scatterers 3. In other words, the metasurface is designed with a dense ratio Dr of two (Dr = 2), for example at 100 GHz. Since sub-terahertz frequencies suffer from high attenuations, this frequency may be chosen but the proposed technique may be applied for any frequencies. As shown in Figure 10, a unit cell 2 comprising two sub-unit cells 4 is designed. The designed sub-unit cells 4, e.g. independent scatterers 3, may be able to have different phase and amplitude responses to provide the proper reflection coefficients according to algorithm requirements of the Delta- Sigma algorithm. The number of sub-unit cells 4 per unit cell 2 and number of independent scatterers 3 per sub-unit cell 4 is only by way of example and does not limit the present disclosure.

The two independent scatterers 3 of each sub-unit cell 4 of the unit cell 2 may be two metal radiators 101. These two metal radiators may operate together and the response of both of them may form the response of the respective sub-unit cell 4. In this way, there is a large degree of freedom to control the response of each sub-unit cell 4 of the unit cell 2. Each of the radiators (i.e. independent scatterers) may have different arc angles (e.g. al, a2), connection lengths (e.g. Li, L2), rotation angles (e.g. 1, P2), inner radius (Ri) and outer radius (Ro). On the right side of Figure 10 the layers 101, 102 and 103 of the designed metasurface are exemplarily shown. The layers 101 and 103 may be made of metal, e.g. copper. The layer 101 may be metals trips for forming the metal radiators and, thus, the independent scatterers 3. The layer 103 may be a ground layer. The layer 102 may be a substrate layer. For example, the substrate layer 102 may be made of a material known as “RT/Duroid 6002". The substrate layer 102 may be made of a dielectric laminate, optionally with a dielectric constant of 2.9, which are low loss materials and provide good (excellent) high frequency performance. That is, the substrate layer 102 may be made of one or more low loss materials (e.g. one or more low loss dielectric materials) providing good (excellent) high frequency performance. With good (excellent) mechanical and electrical properties, the aforementioned materials may be reliable for use in multi-layer board constructions.

Figure 11 shows an example of a super cell 6 comprising four unit cells 2i, 22, 2s and 24. That is, these four unit cells 2i, 22, 2s and 24 form a group of unit cells covering a 2K phase shift. As shown in Figure 11, each unit cell 2 may comprise two sub-unit cells 4, wherein each sub-unit cell 4 comprises two independent scatterers 3. Thus, the super cell 6 of Figure 11 comprises eight sub-unit cells 4i, 42, 4s, 44, 4s, 4e, 4? and 48. The number of unit cells 2 per super cell 6, number of sub-unit cells 4 per unit cell 2, number of independent scatterers 3 per unit cell 2 and number of independent scatterers 3 per sub-unit cell 4 is only by way of example and does not limit the present disclosure. For further details on Figure 11, reference is made to the description ofFigure 10.

The present disclosure has been described in conjunction with various embodiments as examples as well as implementations. However, other variations can be understood and effected by those persons skilled in the art and practicing the claimed matter, from the studies of the drawings, this disclosure and the independent claims. In the claims as well as in the description the word “comprising” does not exclude other elements or steps and the indefinite article “a” or “an” does not exclude a plurality. A single element or other unit may fulfill the functions of several entities or items recited in the claims. The mere fact that certain measures are recited in the mutual different dependent claims does not indicate that a combination of these measures cannot be used in an advantageous implementation.