General Metasurface Synthesis
Based on Susceptibility Tensors
A general method, based on susceptibility tensors, is proposed for the synthesis of metasurfaces transforming arbitrary incident waves into arbitrary reflected and transmitted waves. The proposed method exhibits two advantages: 1) it is inherently vectorial, and therefore better suited for full vectorial (beyond paraxial) electromagnetic problems, 2) it provides closed-form solutions, and is therefore extremely fast. Incidentally, the method reveals that a metasurface is fundamentally capable to transform up to four independent wave triplets (incident, reflected and refracted waves). In addition, the paper provides the closed-form expressions relating the synthesized susceptibilities and the scattering parameters simulated within periodic boundary conditions, which allows one to design the scattering particles realizing the desired susceptibilities. The versatility of the method is illustrated by examples of metasurfaces achieving the following transformations: generalized refraction, reciprocal and non-reciprocal polarization rotation, Bessel vortex beam generation, and orbital angular momentum multiplexing.
Metasurfaces [1, 2, 3] are dimensional reductions of volume metamaterials [4, 5, 6] and functional extensions of frequency selective surfaces . They are composed of two-dimensional arrays of sub-wavelength scattering particles engineered in such a manner that they transform incident waves into desired reflected and transmitted waves. Compared to volume metamaterials, metasurfaces offer the advantage of being lighter, easier to fabricate and less lossy due to their reduced dimensionality, while compared to frequency selective surfaces, they provide greater flexibility and functionalities.
A myriad of metasurfaces have been reported in the literature. For instance, one may mention metasurfaces providing tunable reflection and transmission coefficients , plane wave refraction , single-layer perfect absorption , polarization twisting , and vortex wave generation , and many more metasurface structures and applications are expected to emerge in coming years.
To date, only a small number of metasurface synthesis techniques have been reported [13, 14, 15, 16]. The first two,  and , are based on impedance tensors corresponding to the relations (7). They are similar to the case presented in Sec. IV-C, which represents only a particular case of the general method proposed in the paper. The third technique, , relates the waves reflected and transmitted from the metasurface to the polarizabilities of a single scattering element in the case of a normally incident plane wave. In contrast, this paper deals with wave of arbitrary incident angle and arbitrary type. Finally, the last paper, , proposes a technique called the momentum transformation method, which is a spectral () method, that is particularly suitable for paraxial wave problems. It can also handle full vectorial problems but this involves extra complexity compared to the scalar case.
We propose here an alternative synthesis method, which is inherently vectorial and which may therefore be easier to implement in full vectorial electromagnetic situations. Moreover, this method leads to closed-form solutions, and is hence extremely fast. It describes the metasurface in terms of surface susceptibility tensors in the space domain, where the susceptibility tensors are related through Generalized Sheet Transition Conditions (GSTCs)  to the incident, reflected and transmitted fields around the structure.
The paper is organized as follows. Section II defines the synthesis problem to be solved. Section III explains why conventional textbook boundary conditions are not adequate to handle metasurface problems and establishes, with the help of the Appendix, the metasurface GSTCs for bi-anisotropic metasurfaces. Section IV is the core of paper. It presents the proposed synthesis method, points out the fundamental possibility for a metasurface to simultaneously handle several independent waves and derives closed-form expressions for its susceptibilities for the case of one and two wave transformations. The versatility of the method is illustrated in Sec. V by various examples, where the metasurface is synthesized so as to provide generalized refraction, reciprocal and non-reciprocal polarization rotation, Bessel vortex beam generation, and orbital angular momentum multiplexing. Finally, conclusions are provided in Sec. VI.
Ii Metasurface Synthesis Problem
A metasurface is an electromagnetic two-dimensional structure with sub-wavelength thickness (). The metasurface may be finite, with dimensions , or infinite. It is typically composed of a non-uniform arrangement of planar scattering particles (full or slotted patches, straight or curved strips, various types of crosses, etc.) that transforms incident waves into specified reflected and transmitted waves.
Figure 1 shows the synthesis problem to solve. How can one synthesize a metasurface that transforms an arbitrary specified incident wave, , into an arbitrary specified reflected wave, , and an arbitrary specified transmitted wave, , assuming monochromatic waves? Here the solution will be expressed in terms of the transverse susceptibility tensor functions of , , , and , which represent the electric/magnetic (e/m) transverse polarization responses (first subscript) to transverse electric/magnetic (e/m) field excitations (second subscript).
The synthesis procedure will always yield , , and results, but will not guarantee that these results can be practically implemented using planar scattering particles. For instance, if the susceptibilites exhibit multiple spatial variations per wavelength, it may be difficult or impossible to realize. In such cases, one has to determine whether some features may be neglected or one may have to relax the design constraints (e.g. allow higher reflection or increase the metasurface dimensions).
The complete synthesis of a metasurface typically consists in two steps: 1) determination of the mathematical transfer function of the metasurface producing the specified fields, which is generally a continuous function of the transverse dimensions of the metasurface; 2) discretization of the transfer function obtained in 1) according to a two-dimensional lattice and determination of the scattering particles realizing the corresponding transfer function at each lattice site. Step 2) involves a full-wave parametric analysis of judiciously selected scattering particles, from which magnitude and phase maps are established to find the appropriate particle geometries for building the metasurface using the periodic boundary condition approximation . Since this second step involves scattering parameters, the paper also provides transformation formulas between susceptibilities and scattering parameters to enable the complete synthesis of the metasurface.
Iii Metasurface Boundary Conditions in Terms of Surface Susceptibility Tensors
A metasurface may be considered as an electromagnetic discontinuity in space. Conventional textbook electromagnetic boundary conditions do not apply to such a discontinuity. As was pointed out by Schelkunoff , the mathematical formulation of the conventional boundary conditions is not rigorous in the case of field discontinuities caused by sources, such as surface charges and currents, although it yields satisfactory results away from the discontinuities. Assuming an interface at , the conventional boundary conditions relate the fields at , but fail to describe the field behavior at the discontinuity itself (). This discrepancy is due to the fact that Stokes and Gauss theorems used to derive them assume field continuity in all the regions they apply to, including the interface, whereas the fields may be discontinuous due to the presence of sources. For instance, consider the conventional boundary condition for the normal component of the displacement vector in the presence of surface charges ,
This relation is derived by applying Gauss theorem, , to a volume enclosed by the surface including the interface discontinuity with the normal unit vector to . This theorem rigorously applies only if is continuous inside the entire volume , whereas in the case of a discontinuous , its projection onto is not defined at the interface and application of this theorem is not rigorously correct. Thus, since a metasurface may be modeled by Huygens sources , the correct field behavior on the metasurface cannot be determined using the conventional boundary conditions and rigorous boundary conditions, namely GSTCs, must be applied, as will be done next. It should be noted that, from a physical perspective, a metasurface structure is not a single interface but rather a thin inhomogeneous slab, and may be naturally treated as such. However, it is much simpler to treat the metasurface as a single interface using rigorous GSTCs, which is allowed by the fact that it is electromagnetically thin.
Rigorous GSTCs, treating discontinuities in the sense of distributions, were derived by Idemen . The corresponding relations pertaining to this work, first applied by Kuester et al. to metasurfaces , are derived in Appendix A for the sake of clarity and completeness. They may be written as111Throughout the paper, the medium surrounding the metasurface is assumed to be vacuum, with permittivity and permeability and , respectively. Therefore, for notational compactness, and also to avoid confusion between the subscript ‘0’ meaning ‘vacuum’ or meaning ‘first order discontinuity’ (Appendix A), the subscript ‘0’ for ‘vacuum’ is suppressed everywhere.
In these relations, the terms in the left-hand sides represent the differences between the fields on the two sides of the metasurface, whose cartesian components are defined as
where represents any of the fields , , or , and where the superscripts i, r, and t denote incident, reflected and transmitted fields, and and are the electric and magnetic surface polarization densities, respectively. In the most general case of a bi-anisotropic medium, these densities are related to the acting (or local) fields, and , by [20, 21]
where the terms represent the averaged polarizabilities of a given scatterer, and is the number of scatterers per unit area. The acting fields are, by definition, the average fields on both sides of the surface taking into account the contributions of all the scattering particles (coupling effects) except that of the particle being considered. The contribution of this particle may be modeled by replacing it with a disk of radius encompassing its electric and magnetic current dipoles. Kuester et al. express the fields of this disk as functions of the polarization densities and , from which relations (4) can be rewritten as functions of the average fields. Their relations, with averaged polarizabilities replaced by surface susceptibilities for macroscopic description, read
where the average fields are defined as
where represents either or . The utilization of surface susceptibilities, which represent the actual macroscopic quantities of interest, allows for an easier description of the metasurface than particle averaged polarizabilities and densities in (4).
The surface may be infinite or finite with dimensions . The two problems are automatically solved by specifying the fields , and in (3) and (6) to be of infinite or finite extent in the former and latter cases, respectively. In the finite case, truncation practically corresponds to placing a sheet of absorbing material around the metasurface. This operation neglects diffraction at the edges of the metasurface, as is safely allowed by the fact that a metasurface is generally electrically very large, but properly accounts for the finiteness of the aperture via the GSTCs (2a) and (2b).
Iv Synthesis Method
The proposed synthesis method solves the inverse problem depicted in Fig. 1, where the electromagnetic fields are specified everywhere (for all ) in the plane on both sides of the metasurface and the properties of the metasurface are the unknowns to be determined. We specifically aim at finding the susceptibilities that transform specified incident waves into specified transmitted and reflected waves. The method essentially consists in solving Eqs. (2) for the components of the susceptibility tensors in (5).
The last terms in (2a) and (2b) involve the transverse derivatives of the normal components of the polarization densities, namely and . Solving the inverse problem for non-zero and/or would be quite involved since this would require solving the set of coupled non-homogenous partial differential equations formed by (2a) and (2b) with nonzero and . Although such a problem could be generally addressed by means of numerical analysis, we enforce here , which will lead to convenient closed-form solutions for the susceptibilities222This restriction may limit the physical realizability of the metasurface in some cases, in the sense that the corresponding synthesized susceptibilities might be excessively difficult to realize with practical scattering particles. In such cases, the restriction might be removed without changing the main spirit of the method but at the risk of losing the closed-form nature of the solution.. As shall be seen next, this restriction still allows the metasurface to realize a large number of operations, given the large number of degrees of freedom provided by combinations of its bi-anisotropic susceptibility tensor components.
The method needs considering only (2a) and (2b) as these two equations involve all the transverse field components, which is sufficient to completely describe the fields at each side of the metasurface according to the uniqueness theorem. These two equations, with , represent a set a four linear equations relating the transverse electric and magnetic fields to the effective surface susceptibilities. Thus, the solution of the inverse problem will consist in determining four transverse effective susceptibility tensors in (5).
However, the proposed method can also handle the most general case including normal components of the polarization densities, which may be of practical interest (e.g. conducting rings in the metasurface plane, producing contributions). To show this, Sec. IV-E presents the developments required to synthesize metasurfaces involving nonzero and .
Iv-B General Solution for Surface Susceptibilities
As announced in Sec. IV-A, the four susceptibility tensors in (5) are restricted to their four transverse components, and these components will be determined for the specified fields using (2a) and (2b) with , i.e., using the notation in (3) and (6),
Assuming single incident, reflected and transmitted waves (only one wave of each of the three types), the system (7) contains 4 equations for a total number of 16 unknown susceptibility components. It is thus underdetermined as such, and it can be solved only by restricting the number of independent susceptibilities to 4. This single-transformation underdetermination of (7) reveals two important facts: i) Many different combinations of susceptibilities produce the same fields; ii) A metasurface has the fundamental capability to simultaneously manipulate several linearily independent incident, reflected and transmitted waves. Specifically, a metasurface, as defined by (7b), can in principle manipulate up to 4 sets of incident, reflected and transmitted waves. If () waves are to be manipulated, corresponding to independent equations obtained by writing the 4 equations in (7b) for each of the field sets (, representing either or ), (4,8,12,16) susceptibilities have to be specified.
Two approaches may be considered to reduce the number of independent unknown susceptibilities when . A first approach could consist in using more than (4,8,12) susceptibilities but enforcing relationships between some of them to ensure a maximum of independent unknowns. For example, the conditions of reciprocity and losslessness would be a possible way to link some susceptibilities together, if this is compatible with design specifications. According to Kong  and Lindell , the conditions for reciprocity and losslessness are
respectively, where the superscripts T and denote the matrix transpose and complex conjugate operations, respectively. Enforcing conditions between susceptibilities also enforces conditions on the fields on both sides of the metasurface. Therefore, this approach restricts the diversity of electromagnetic transformations achievable with the metasurface.
A second approach, providing a more general synthesis method for quasi-arbitrary electromagnetic transformations, is then preferred. This approach consists in selecting only susceptibility tensor components in each of the equations included in (7). The number of possible susceptibilities in each equation is given by , where and . Therefore, for , and the total number of different combinations for the equations is . By the same token, we respectively have for and the following total number of different combinations and . Among these combinations, those that do not satisfy (8) naturally correspond to non-reciprocal and/or lossy designs. It is obviously impossible to cover these huge numbers of synthesis combination sets for . We will therefore next, without loss of generality, restrict our attention to the cases of single (Sec. IV-C) and double (Sec. IV-D) transformations with mono-anisotropic metasurfaces. The solutions for bi-anisotropic, triple-wave and quadruple-wave metasurfaces can be obtained by following exactly the same procedure.
Note that these considerations hold in the most general case of transformations where the amplitude, phase and polarization of the incident field are all modified by the metasurface. Under such conditions, the system in (7) can indeed handle up to independent wave triplets. In the particular case of single-triplet transformation, only 4 susceptibilites (2 electric and 2 magnetic) are generally required, as will be shown shortly. However, as will be seen in the example of generalized refraction in Sec. V-A, depending on the transformation and choice of susceptibilities, only 2 susceptibilities (1 electric and 1 magnetic) may be sufficient. In general, this factor 2 reduction in required susceptibilities indicates that the number of possible transformations doubles, i.e. .
What has been described so far in this section represents the first step of the synthesis procedure. As mentioned in Sec. II, the second step consists in determining the scattering particles realizing the transfer function corresponding to the synthesized susceptibilities. In this second step, one computes the full-wave scattering parameters for an isolated unit cell within 2D periodic boundary conditions, where periodicity is an approximation of typically slowly varying scattering elements in the plane of the metasurface [18, 22, 23, 9]. The periodic boundary conditions in full-wave analysis are generally restricted to rectilinearly propagating waves. Now, the prescribed waves may change directions at the metasurface (e.g. case of generalized refraction, Sec. V-A). In such cases, “rectilinear” periodic boundary conditions cannot directly describe the physics of the problem. However, the results they provide correspond to a rigorous mapping with the physical problem, and they may thus be rigorously used in the synthesis, as will be illustrated in Sec. V-A.
Iv-C Single Transformation
We consider here the problem of single () transformation [only one specified wave triplet: (, , )] for a mono-anisotropic () and uniaxial (), and hence non-gyrotropic and reciprocal, metasurface. Solving (7) under these conditions yields the following simple relations for the remaining 4 susceptibilities
By synthesis, a metasurface with the susceptibilities given by (9) will produce exactly the specified reflected and transmitted transverse components of the fields when the metasurface is illuminated by the specified incident field. Since the longitudinal fields are completely determined from the transverse components, according to the uniqueness theorem, the complete specified electromagnetic fields are exactly generated by the metasurface.
Consistency with Maxwell equations can be easily verified. Consider for instance (2c) along with the relation ,
which upon substitution of (9) becomes
This equation represents a relation between linear combinations of the longitudinal electric fields and derivatives of the transverse magnetic fields of the incident (ki), reflected (kr) and transmitted (kt) waves. From linearity, and subsequent superposition, these equations may be decomposed as
which is nothing but the projection of Maxwell-Ampère equation upon the direction. This shows that the longitudinal fields are well defined with the relations (9) in accordance with the uniqueness theorem.
We are now interested in finding the relationship linking the transmitted field to the incident field and the susceptibilities. In order to simplify the problem, we consider here the case of a reflection-less metasurface. Inserting (3) and (6) with () into (9) and solving for the transmitted components of the fields yields
These relations show how each of the transmitted field components depend on their incident field counterparts and orthogonal duals, e.g. , etc. They have to be considered after the susceptibilities (9) have been synthesized for given specifications to determine whether they may be realized by a passive metasurface ( and ), or require active elements.
The susceptibilities in (9) represent the synthesis (inverse problem) results of the proposed method while Eqs. (14) express the transmitted field components in terms of these susceptibilities (direct problem). We now need to establish the relationships existing between the susceptibilities and the scattering parameters in order to enable the second step of the synthesis (see last paragraph in Sec. II). The forthcoming methodology for single transformation is analogous to that proposed in [8, 9, 13], while the corresponding methodologies for multiple transformation, to be presented in the next subsection, are more general.
In the plane wave approximation, which is naturally valid when the source of the incident wave is far enough from the metasurface, the response of each scattering particle may be expressed in terms of its reflection and transmission coefficients [18, 22, 23]. Since according to (14), the pairs (, ) and (, ) are proportional to their incident counterparts and orthogonal duals only, the problem splits into an -polarized incident plane wave problem and a -polarized incident plane wave problem, whose fields at normal incidence are respectively given by
where and () represent reflection and transmission coefficients, respectively333The waves in (15) and (16) are defined as rectilinear (i.e. they do not change direction at the metasurface) for consistency with periodic boundary conditions to be used in full-wave simulations for the second step of the synthesis (see comment at the end of the last paragraph of Sec. IV-B).. Inserting (15) and (16) into (7) with the four non-zero susceptibilities given in (9) leads to the transmission and reflection coefficients
. These relations may be used in the second step of the synthesis to determine the scattering parameters corresponding to the synthesized susceptibilities. Solving (17) and (18) for the susceptibilities yields
In (18) and (19), the reflection and transmission coefficients are associated with scattering parameters with accounting for the two ports (incident and transmitted waves) and two polarization ( and ). Specifically, assigning ports 1, 2, 3 and 4 to -polarized input, -polarized input, -polarized output and -polarized output, respectively, one has , , and , while the other 12 scattering parameters are not required since the chosen tensors are uniaxial so that the metasurface is not gyrotropic (i.e. does not involve transformations between -polarized and -polarized waves).
Examples of these transformations between the synthesized susceptibilities and the scattering parameters, as given by equations (17) and (18), are presented in Sec. V-A and V-C. As will be shown in Sec. V, the synthesized susceptibilities generally have both real and imaginary parts. Assuming the convention , the imaginary and real parts of the susceptibilities may be associated with gain, when , and loss, when , however, this is not generally true, as compensations may exist between the imaginary parts of different susceptibility terms. It is therefore generally necessary to explicitly compute the refractive index corresponding to the considered susceptibility tensors to determine whether there is really loss or gain. For example, the metasurface described by relations (9) has the property of birefringence, with refractive indices given by
An example of such compensation will be shown in Sec. V-B, where the purely imaginary electric and magnetic susceptibilities in (42) actually compensate each other so that the resulting refractive indices (43) are purely real.
In order to simplify the metasurface unit cell design procedure, which is usually performed via full-wave simulation, one may consider “ideal” unit cells, i.e. unit cells made of lossless dielectric substrates and PEC metallic patterns. Applying relations (19) to compute the susceptibilities corresponding to such “ideal” unit cells will necessarily yield purely real susceptibilities since Eqs. (19) constitute two pairs of orthogonal TEM waves. As a consequence, the exact complex susceptibilities can not be realized, which will lead to results diverging from the specified transformation. A solution to minimize discrepancies between the specified response and the approximated response considering only real susceptibilities is to set the imaginary parts of the susceptibilities to zero while optimizing their real parts so that the response of the metasurface follows the specified response as closely as possible. This is what has been done in , where the optimization procedure consists in minimizing the cost function
where and are found from (17) (or (18) for -polarized waves) using exact synthesized susceptibilities. The terms and are obtained from the same equations but with susceptibilities that are now purely real and have to be optimized to minimize (21).
Iv-D Double Transformation
We now consider the problem of double () transformation [two specified wave triplets: (, , ) and (, , )] for a mono-anisotropic () but not uniaxial and hence gyrotropic metasurface. Solving the equations (7) for the two sets of waves under these conditions with the 2 corresponding susceptibilities per equation yields the following relations for the 8 remaining susceptibilities
where the subscripts and stand for the first and the second wave triplet transformation, respectively.
Following a similar procedure as in Sec. IV-C but assuming that the incident wave given in (15a) induces not only a co-polarized but also a cross-polarized reflected ( and ) and transmitted waves ( and )444For instance, the second equation in (15a) becomes .. And similarly for the incident wave (16a) with the coefficients and . Finally, the double-transformation relations between the reflection coefficients and susceptibilities read
In these relations, assigning ports 1, 2, 3 and 4 to -polarized input, -polarized input, -polarized output and -polarized output, respectively, one has , , and .
Iv-E Solutions for Normal Polarization Densities
Solving the general bi-anisotropic system of equations (2) in the case of full electric and magnetic susceptibility tensors (i.e. nonzero and ) leads to a total number of 36 unknowns for only 6 equations. This corresponds to a heavily under-determined system for single-wave transformation. This means that the system has effectively the capability to handle up to 6 wave triplets, or transformations, in the most general case or more depending on the choice of susceptibilities and the specified transformations (e.g. no rotation of polarization). For single-wave transformation, only 6 (or even less) independent susceptibilities may be set to nonzero values, since this would lead to an exactly determined system of equal number of equations and unknowns.