1. Introduction

The interaction of electromagnetic radiation with anisotropic media is an essential part of modern electrodynamics that has sparked many ideas on how polarization states can be manipulated with different materials. Whether a material is isotropic, anisotropic or even chiral is ultimately determined by its symmetry properties. These can have their origin either in molecular properties or lattice properties of periodic media such as crystals. Of particular interest are materials with artificial subwavelength structures in various arrangements, known as metamaterials [1]. They allow anisotropy to be designed freely from the geometry, material and distribution of their constituent subwavelength structures [2, 3]. In recent years the two-dimensional versions of metamaterials, so called metasurfaces [4, 5], manifested themselves as the most widely used architecture [6, 7].

Anisotropic metasurfaces can be engineered to manipulate polarization generating effects such as asymmetric transmission [8–10], dichroism [11, 12] or optical activity [13, 14]. If the geometry of the structures used is such that they cannot be superposed with their mirror, image they become chiral and can produce a chiral polarization response in turn [15]. To be geometrically chiral, structures have to be three-dimensional [16–19]. Nevertheless, two-dimensional structure designs have been demonstrated, which can also achieve a chiral polarization response [20, 21]. A prominent example are resonant particles such as asymmetric split ring resonators [22, 21].Yet another approach uses bianisotropic bilayers of metasurfaces where the structures are neither chiral nor bianisotropic themselves. Here, bianisotropy is induced through near-field coupling between the layers and can create or enhance chiral polarization effects [10, 23–25]. Following this approach achiral particles like wires [26, 27] or crosses [28, 29] can be rotated with respect to each other to achieve chiral polarization effects [30].

Taking a similar path, it was shown by Zhao et.al. [31] that multiple layers of successively rotated nano-wires can form a so called twisted metamaterial with a chiral polarization response in the absence of near-field coupling. In addition, their work demonstrated that lateral layer alignment during fabrication could be ignored in this type of stacked metasurface [26]. This made apparent that stacking different geometries is a promising route towards designing polarization properties using only basic structure types (e.g. wires, Ls or crosses) and, thus, less challenging fabrication schemes. However, when combining metasurfaces vertically, it is frequently overlooked that spacer layers do not only separate the functional layers of the stack but determine their physical interaction as well. Here, we aim to describe spacer layers as an extra degree of freedom in the design of stacked metasurfaces.

Indeed, the understanding of inter-layer interactions controlled by spacer layers is of great importance from the point of homogenization. It was shown that a periodic metasurface can be considered to be an effective medium if and only if its far-field response is governed by a fundamental mode [32, 33]. For multiple layers of metasurfaces this means that the inter-layer interaction in the stack is solely determined via the fundamental modes of the layers [34]. This differs from the before mentioned case of closely stacked bilayers. In that case, due to the near-field coupling their inter-layer interaction is also influenced by higher order evanescent modes [35, 36]. On the other hand, if distances between metasurface layers are large enough such that all higher order evanescent modes have decayed, interaction on the far-field level via fundamental modes can be assumed [36, 37]. The resulting stack can then be described by the properties of its constituent layers [34]. This, in turn, gives an intuitive perspective on their physical role in the stack, including the influence of spacer layers on anisotropic polarization effects. Ideologically and from a point analytic modeling, this far-field stacking of multiple layers is similar to the simplified analytical description of nano-structures using laterally coupled dipoles [38] or the coupling of resonant modes between periodic metasurfaces [39].

In this paper we consider a twisted metamaterial consisting of a stack of two nano-wire metasurfaces separated by a dielectric spacer, where the wires of one layer are rotated with respect to the ones of the other. We investigate the influence of the spacer layer on the polarization response of the stack and demonstrate how semi-analytics can give a precise understanding of Fabry-Perot-type interactions between stacked metasurfaces. We start with an analysis of the stack’s polarization properties based on the symmetry of its constituent layers. We demonstrate how to derive these properties by analyzing the structure of scattering matrices (S-matrices) describing the system (Sec. 2). Based on these considerations we introduce and investigate a fabricated stack that was designed to have pronounced asymmetric transmission, a strong difference in cross-polarization for orthogonal input polarization. Two versions of the stack are shown, with identical order and type of layers but different spacer thickness (Sec. 3). One refers to the case of far-field coupling and is analyzed using a semi-analytic stacking algorithm (SASA) considering only the fundamental mode of each layer. For comparison, the other incorporates near-field coupling. The focus of our analysis lies on the far-field coupled version to showcase the benefits of a semi-analytic approach. Using SASA we reveal and discuss the role of Fabry-Perot-type interactions for tailoring the polarization properties of the device.

 

Fig. 1 Rendered sketch of a stack of gold nano-wire metasurfaces embedded in glass.

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2. Deducing polarization from S-matrix structure

We consider the case of two metasurfaces stacked along the z-axis, symmetrically embedded in glass, separated by a spacer of thickness d_sp, and with a cladding layer of thickness d_c

as a cover. The lower metasurface is comprised of parallel nano-wires, measuring $w\times l\times h$, aligned to the y-axis, and arranged in a square lattice. The upper metasurface is of the same type as the lower one but its wires are rotated as shown in Fig. 1. To describe the response of the stack and its constituent layers for excitation with normal incident linearly polarized light we make use of optical scattering matrices (S-matrices). An S-matrix is a 2-by-2 block matrix that contains the Jones-matrices $\widehat{T}$ for forward (f) and backward (b) transmission on its diagonal and the forward and backward reflection matrices $\widehat{R}$ on its off-diagonal. They have the meaning of Jones matrices in reflection. Thus, an arbitrary S-matrix S is defined as [34]

Each Jones-matrix contains the four complex transmission or reflection coefficients $T_{xx}$, $T_{xy}$, $T_{yx}$, and $T_{yy}$ or $R_{xx}$, $R_{xy}$, $R_{yx}$, and $R_{yy}$, respectively. Here, the right index of each index pair denotes the ingoing polarization state and the left index the outgoing one.

Our formulation relies on the condition that coupling between layers solely happens in the far-field via a propagating fundamental mode [34]. For periodic metasurfaces with period Λ this is represented by the zeroth diffraction order, where all higher orders decay evanescently. Generally, this is known as the fundamental mode approximation (FMA) [32, 35, 36]. If the FMA is valid for a metasurface, its S-matrix takes the form of a 4-by-4 matrix with two-dimensional block matrices. Furthermore, if all layers of a stack can be considered to be in the FMA, the S-matrix of the entire system can be calculated analytically as a combination of its constituent S-matrices using Redheffer’s starproduct $\cdot *\cdot$ [40, 41]. As long as the FMA is valid any type of metasurface can be described and stacked in this way using either analytic, numerical or measured S-matrices. Utilizing this as a semi-analytic stacking algorithm (SASA) we can model stacks with an arbitrary number of layers of almost any type. In the following, we will use SASA to analyze the polarization behavior of the stack qualitatively from the structure of its S-matrices using only symbolic representations of their coefficients.

As the simplest part of the stack, the S-matrix of a dielectric spacer with refractive index n can be written as

There are serval things to note. First, as intuition would tell us, all transmission coefficients are multiplied by an identical phase term such that reciprocity is not affected [42]. Second, reflection coefficients add twice the phase depending on the side of incidence. Additionally, the overall structure of the S-matrix will not be changed by the spacer S-matrix because of the block-wise application of the phase terms.

The S-matrices of two wire-based metasurfaces from Fig. 1 can be constructed analytically. The only assumptions we need are the following: the particle geometry is identical in both metasurfaces, where one has wires aligned to the y-axis, the other wires rotated by an angle α about the z-axis, and both are symmetrically embedded. Similar to the discussion of unit cell symmetries and the resulting Jones matrix symmetries in references [2] and [34] we can derive each of our S-matrices by considering the polarization properties of nano-wires. We know that a nano-wire will have a different polarization response depending along which axis it is excited. It is also clear from the metasurfaces’ symmetric embedding that its response is reciprocal and, thus, its S-matrix is symmetric [43]. Accordingly, it is straight forward to write the S-matrix of the wires S_w as

For readability we chose placeholder variables defined by ${\tilde{T}}_{\text{xx}}=T_{\mathrm{x}}\stackrel{2}{\cos }\alpha +T_{\mathrm{y}}\stackrel{2}{\sin }\alpha$, ${\tilde{T}}_{\text{yy}}=T_{\mathrm{y}}\stackrel{2}{\cos }\alpha +T_{\mathrm{x}}\stackrel{2}{\sin }\alpha$, ${\tilde{T}}_{\text{xy}}=\left(T_{\mathrm{y}}-T_{\mathrm{x}}\right)\text{cos }\alpha \text{sin }\alpha$, ${\tilde{R}}_{\text{xx}}=R_{\mathrm{x}}\stackrel{2}{\cos }\alpha +R_{\mathrm{y}}\stackrel{2}{\sin }\alpha$, ${\tilde{R}}_{\text{yy}}=R_{\mathrm{y}}\stackrel{2}{\cos }\alpha +R_{\mathrm{x}}\stackrel{2}{\sin }\alpha$, and ${\tilde{R}}_{\text{xy}}=\left(R_{\mathrm{y}}-R_{\mathrm{x}}\right)\text{cos }\alpha \text{sin }\alpha$. The structure of ${\tilde{S}}_w$ already shows that rotated wires may produce cross-polarization. However, this alone does not produce asymmetric transmission which is defined for reciprocal media and incident x-polarization (analogously for y-polarization) as the difference of the cross-polarization components in forward (or backward) direction [2],

In case of eq. (5) the cross-polarization terms are identical (${\tilde{T}}_{\text{xy}}={\tilde{T}}_{\text{yx}}$ and ${\tilde{R}}_{\text{xy}}={\tilde{R}}_{\text{yx}}$) and Δ_x vanishes.

Eqs. (2), (4), and (5) give us the layers of the desired stack of rotated nano-wires on top of axis-aligned nano-wires. Performing the starproduct on the three S-matrices consecutively results in

The explicit expressions of the resulting coefficients are given in the appendix and reveal that ${\overline{T}}_{\text{xy}}\ne {\overline{T}}_{\text{yx}}$, i.e. asymmetric transmission Δ_x emerges from the stacking of the individual S-matrices. It is noteworthy that we can determine the general polarization behavior of the complete stack only from the symmetry properties of the individual layers.

 

Fig. 2 (a) SEM picture of a FIB cut revealing the metasurface layers of the stack. The nano-particles in this image were colored golden during postprocessing to enhance visibility. (b) SEM image of the lower layer showing axis-aligned nano-wires. (c) SEM image of the upper layer showing nano-wires rotated counterclockwise by ${60}^{\circ }$.

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Fig. 3 Experimental and simulation results of the stack in Jones matrix form: a) $T_{\mathbf{xx}}$, b) $T_{\mathbf{xy}}$, c) $T_{\mathbf{yx}}$, d) $T_{\mathbf{yy}}$. Blue curves show transmittance and orange ones phase. Measured data is represented by solid and dashed lines showing results for the 345 nm stack and 40 nm stack, respectively. Results obtained via SASA are marked with crosses and those from rigorous FMM calculations with circles.

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Fig. 4 Sketch of the stack construction with S-matrices using eq. (9). Dashed lines mark the interfaces of the stack to the surrounding medium. Arrows point in the direction of (forward) propagation along the z-axis. While spacer (spac.), and cladding (clad.) have defined thicknesses, the two metasurface layers (MS${ }_{\mathbf{l}}$, MS${ }_{\mathbf{u}}$) are assumed to be infinitely thin.

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3. Analysis of a fabricated stack by semi-analytic modeling

We fabricated a stack resembling the one shown in Fig. 1 and analyzed it both experimentally and employing SASA. For simplicity we chose equal periods with $\mathrm{\Lambda }=400\text{nm}$ and aligned unit cells for both metasurfaces. The rotation angle of the upper layer metasurface’s wires was set to α = 60°. During fabrication of each metasurface we applied a two-step electron beam lithography process. They were structured by exposure of a two layer electron beam resist (150 nm Allresist ARP617.08 and on top of that 100 nm Allresist ARP6200.4) with a variably shaped electron beam (Vistec, SB 350). This was followed by chemical development and coating by gold evaporation. Finally, we applied a lift-off process to remove the resist. During this process we produced two fields of the same metasurface. The spacer layer was spin-coated (Futurrex IC1-200) on top of both fields, tempered at 200°C for half an hour and etched to the desired spacer height. We produced two versions of the stack: spacer thickness $d_{\text{sp}}=$40 nm on one field and $d_{\text{sp}}=$ 345 nm on the other. The former showcases the effect of evanescent coupling as opposed to the latter, coupling only in the far-field. The second metasurface was fabricated with the same electron beam lithography process on top of the respective spacer. In this way, we made sure that both versions were comparable with nearly identical structures. Finally, a cladding layer was spin-coated to $d_{\mathrm{c}}=$ 900 nm. In total the field size of each fabricated sample was 2 × 2 mm².

A scanning electron microscope (SEM) image of the resulting stack with $d_{\text{sp}}=$ 345 nm is shown in Fig. 2(a), where parts of the layers were removed with a focused ion beam (FIB) to reveal the structures underneath. Average dimensions of the nano-wires of both layers, Figs. 2(b) and (c), are almost identical, with: $l_{\mathrm{l}}=$ 235 nm, $w_{\mathrm{l}}=$ 75 nm, $h_{\mathrm{l}}=$ 50 nm, $l_{\mathrm{u}}=$ 225 nm, $w_{\mathrm{u}}=$ 75 nm, and $h_{\mathrm{u}}=$ 75 nm, with l, w, and h defined in Fig. 1. The subscripts l and u denote the lower and upper layer, respectively.

We characterized the stack in a wavelength range from 600 nm to 1620 nm using an interferometric setup which measures both the transmitted intensity (transmittance) and phase in a linear polarization basis [17, 44]. The diameter of the characterization beam on the sample was 2 mm, making sure that it was fully illuminated and no finite-size effects occurred. The results for both variants of the stack are plotted in Fig. 3, arranged in the form of a Jones matrix. The co-polarization components for x- and y-polarization, Figs. 3(a) and (d), vary only slightly in transmittance and are almost identical in phase. The difference between the respective cross-polarization components, Figs. 3(c) and (b), is more noticeable since they have much lower overall transmittance. In general, the functionality of both versions of the stack is similar. However, the peak in $T_{yx}$, Fig. 3(c), exhibits a redshift of 150 nm for the thin spacer stack. This indicates near-field coupling between the metasurfaces of the $d_{\mathrm{c}}=$ 40 nm stack.

To investigate the dependence of the stack’s polarization behavior on the spacer we developed a model of the $d_{\text{sp}}=$ 345 nm version as a sequence of layers similar to the semi-analytic model eq. (7) using SASA. Now, interfaces to substrate and cladding of the stack have to be included, as shown in Fig. 4. Since the sample was fabricated and characterized on a glass waver, the substrate was assumed to be infinite. Both the spacer and the cladding layer on top of the stack were modeled analytically using equation (2). Likewise, the interfaces were calculated using the interface S-matrix S_i [34],

 

Fig. 5 (a)-(b) Linear asymmetric transmission (a) and ellipticity (b) derived from measurement (solid lines) and SASA results (dashed lines). Red indicates x-polarized and blue y-polarized incident light. (c)-(e) Asymmetric transmission and ellipticity scanned over the spacer thickness from 40 nm to 1000 nm. (c) Asymmetric transmission for x-polarization (y-polarization as its negative counterpart was omitted); (d) and (e) display ellipticity for x- and y-polarization. The red line in each surface plot marks a spacer thickness of $d_{\mathbf{sp}}=$ 345 nm, corresponding to solid lines in (a) and (b).

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For quantitative modeling, the S-matrix coefficients of both metasurfaces S_l and S_u were determined numerically as single layers, using a self-implemented Fourier-Modal-Method (FMM) [45, 46] and the geometric parameters from SEM measurement. Based on a beforehand done convergence test the FMM was truncated at $15$

Fourier orders (in x- and y-direction). In case of the upper metasurface the nano-wires were rotated individually in their unit cell in adaption to the fabricated sample. To account for the corner rounding of the fabricated structures a curvature radius of 25 nm was assumed for the corners of our model particles. As material parameters we used measured ellipsometric data of evaporated gold and spin-on glass as they result from our fabrication process [47].

In addition to SASA a rigorous FMM of the full stack was performed to compare and determined the correctness of the semi-analytical model. Since a rigorous FMM takes all evanescent and propagating orders into account up to a set truncation limit (here: $15$ Fourier orders) [41, 46], a direct comparison to SASA will help to indicate the validity of the FMA. As a side note, the computation times of the rigorous FMM and SASA are compared in the appendix.

The resulting transmittance and phase plots of both methods are shown together with the experimental results in Fig. 3. SASA and FMM results coincide almost perfectly both in transmittance and phase, indicating the validity of the FMA. Notably, all the main features of the measured curves were reproduced and the overall agreement between experimental and theoretical results is very good.

Next, in order to analyze the polarization behavior of the stack we derived its asymmetric transmission Δ_x, eq. (6), and ellipticity ϵ in forward transmission. For incident x-polarization (analogously for y-polarization) ϵ_x is defined as the ratio of the major and minor half axis of the polarization ellipse,

We note that asymmetric transmission, Fig. 5(a), is strongest in the spectral range of the peak around ${\lambda }_0=$ 950 nm, reaching up to ± 12 %. In the same spectral region, we see a strong conversion from x-polarized to elliptically polarized light close to being circular, Fig. 5(b). The ellipticity for incident y-polarized light is more shallow and does not go beyond $\pm 0.5$. This is explained by the presence of asymmetric transmission giving light a different twist depending on the incident polarization.

To further clarify the role of the spacer for asymmetric transmission we varied its thickness from 40 nm to 1000 nm in 1 nm steps using SASA. Applying the associativity of the starproduct [41], such that

This enabled us to determine Δ_x and ${ϵ}_{x/y}$ as a function of d_sp, as shown in Fig. 5(c)-(e). The resulting Fabry-Perot type patterns show that the polarization characteristic of this system is highly sensitive with respect to the spacer thickness. Both the emerging polarization states and the dispersion of asymmetric transmission change as the Fabry-Perot condition changes with the spacer thickness. Moreover, the SASA result reveals that the ellipticity in y-polarization is not deviating from its flat characteristic from 1000 nm to longer wavelengths for different spacer thicknesses. We can deduce that in this spectral region y-polarized light is converted to a slightly elliptical state mostly determined by the geometry of the stack. However, it should also be noted that in this particular region both cross-polarization components of the stack almost vanish.

Our analysis of the stack shows that the resulting asymmetric transmission Δ_x is sensitive to the chosen spacer thickness d_sp. Indeed, a different choice of d_sp could almost result in Δ_x vanishing. Thus, when designing chiral stacks of this type it is not sufficient to define the geometry of the structure but necessary to also optimize the separation between the metasurfaces. In conclusion, the spacer thickness of complex metasurface stacks represents another degree of freedom in the design of sophisticated layered media. Besides simply controlling the occurrence of Fabry-Perot resonances it can be utilized to enhance otherwise negligible polarization effects. It is also imaginable to overlap Fabry-Perot resonances with other resonant effects to modify a stack’s response for different functionalities.

4. Summary

Using a scattering matrix based approach we derived the polarization behavior of a twisted metamaterial from basic symmetry considerations. The twisted metamaterial consisted of two stacked gold nano-wire metasurfaces with a dielectric spacer in between, where the wires of the upper metasurface were rotated with respect to the lower ones. At first, we derived the stack’s polarization properties without specifying its constituent materials or their exact geometric measures, demonstrating how to construct polarization effects from general symmetry considerations. Viewing the stack layer by layer we could show analytically that it exhibits asymmetric transmission even though the symmetry of each layer alone could not produce this effect. The predicted polarization behavior was realized in two fabricated variants of the modeled stack with different spacer thickness. One targeted the regime of near-field coupling between the metasurfaces and the other coupling in the far-field. The latter had the advantage of being accessible by fast semi-analytic modeling, whereas the former was used to showcase the difference and similarities between the two cases. Our results showed almost perfect agreement between full-wave simulations and our semi-analytic stacking algorithm, and a very good match to measurements. It was thus possible to numerically investigate the physical role of the spacer, revealing a Fabry-Perot-type modal resonance which can be spectrally tailored. This showed a distinct influence of the spacer height on the polarization characteristics of the stack. With this we demonstrated the important role of spacers in the design of stacked metamaterials which are often considered to be nothing but placeholder between functional layers. With the ability of using the spacer thickness as a free design parameter far-field coupled stacks of arbitrary metasurfaces represent ideal candidates for easily designable sophisticated layered media.