Abstract
The circular dichroism (CD) of a material is the difference in optical absorption under left and rightcircularly polarized illumination. It is crucial for a number of applications, from molecular sensing to the design of circularly polarized thermal light sources. The CD in natural materials is typically weak, leading to the exploitation of artificial chiral materials. Layered chiral woodpile structures are well known to boost chirooptical effects when realized as a photonic crystal or an optical metamaterial. We here demonstrate that light scattering at a chiral plasmonic woodpile, which is structured on the order of the wavelength of the light, can be well understood by considering the fundamental evanescent Floquet states within the structure. In particular, we report a broadband circular polarization bandgap in the complex band structure of various plasmonic woodpiles that spans the optical transparency window of the atmosphere between 3 and 4 \(\upmu\)m and leads to an average CD of up to 90% within this spectral range. Our findings could pave the way for an ultrabroadband circularly polarized thermal source.
1 Introduction
The manipulation of light through optical elements such as lenses, color filters, and polarizers goes back to ancient times and can ubiquitously be found in the living world [1, 2]. The limits that naturally available materials impose on light manipulation can be overcome by nanostructuring matter in form of for example photonic crystals, dielectric geometries that are structured on the order of the wavelength of the light and lead to interference effects [3, 4]. A different strategy to obtain new optical materials employs plasmonic metals structured on a deeply subwavelength scale, socalled metamaterials [5]. A vast number of chiral photonic crystal and metamaterial designs [6] have been suggested to yield customdesigned chirooptical effects, such as strong circular dichroism [7,8,9,10,11,12], optical activity [13,14,15,16,17,18], and orbital angular momentum generation [19,20,21]. Next to a number of applications in sensing, catalysis, and chiral light generation, when realized on a micrometer length scale, these chiral geometries can be engineered to yield circularly polarized thermal emission by application of Kirchhoff’s law [22,23,24,25,26,27,28,29,30,31,32].
A particularly promising and relatively easy to manufacture (down to submicrometer length scales) geometry is the chiral woodpile [34,35,36,37,38] as illustrated in Fig. 1. When realized as a classical deeply subwavelength metamaterial, the chiral woodpile, which is evidently not based on local metaatoms, does not show a strong chirooptical response due to the mismatch between the screwaxis pitch and the vacuum wavelength. A hexagonal chiral woodpile realized as a lossy photonic crystal with big index contrast (using a semiconductor with \(\varepsilon \,{=}\,8.9\)), on the other hand, is predicted to produce circularly polarized thermal emission within a broad band (gaptomidgap ratio \(\Delta \omega /\langle \omega \rangle \approx 1/6\)) [22]. While a more sophisticated fully threedimensional geometry with \(\varepsilon \,{=}\,12\) is predicted to improve the bandwidth to a gaptomidgap ratio of 1/3 [12], we here instead consider a plasmonic hexagonal woodpile structured on the order of the wavelength that leads to a strongly circularly polarized thermal emission within a band with a gaptomidgap ratio \({\gtrsim }\,2/7\), focusing on the optical transparency window of the atmosphere between 3 and 4\(\upmu\)m. A similar structure has been previously considered to engineer linearly polarized thermal emission [36]. Since the spatial periodicity of such a plasmonic crystal (PC) is on the order of the wavelength, standard metamaterial homogenization techniques cannot be applied. Looking at the field patterns can shed some light on the chiral plasmonic excitations, as for example done for a helix PC [39].
The simulated scattering of an outside source, however, always corresponds to an apriory unknown superposition of selfconsistent modal solutions within the PC. A physically inspired set of such modal solutions is provided through evanescent Floquet states [40,41,42]. These states can be thought of as a natural complex generalization of the Bloch modes discussed in the canonical band structure picture employed in the photonic crystal theory [3]. They thus allow to account for base materials with general optical permittivity and permeability and span the solution space of the monochromatic Maxwell equations within finite slablike geometries with periodicity. While the calculation of evanescent Floquet modes is not implemented in standard simulation tools based on the finite element or finite difference methods, they can be naturally obtained through a planewave approach [40] for photonic crystals. For plasmonic materials, such an approach is, however, very inefficient as the generally poor convergence behavior of the planewave basis becomes a particularly big problem at metaldielectric interfaces [41], where the permittivity changes its sign. In simple geometries, which are homogeneous in the propagation direction, such as the lamellar grating [43], the fishnet structure [42], or hyperbolic aligned wire media [44, 45], the computation simplifies to an essentially one or twodimensional problem. More generally, in a geometry that is made by a number of slices that are individually homogeneous in the propagation direction, the evanescent Floquet modes are efficiently calculated as the eigensolutions of the corresponding transfer matrix through the unit cell [46,47,48,49,50].
We here show that the fundamental Floquet modes responsible for propagating energy through (and absorbing it within) a single lamellar grating layer are sufficient to calculate the two fundamental Floquet modes of a hexagonal chiral woodpile made of a sequence of equal lamellar grating layers, copied through a \(C_{6_2}\) screw rotation, as illustrated in Fig. 1d. These fundamental Floquet states of the chiral woodpile in turn describe the scattering physics well and reveal the origin of a broad circularly polarized stop band in the absorption spectrum, which is induced through a circularly polarized pseudobandgap. We use these two Floquet modes as a scattering basis within the fundamental Bragg order to efficiently optimize the broadband circular dichroism and derive general design principles based on the thickness of the dielectric region \(d_1\) and the chiral pitch c only (Fig. 1). We show that the obtained absorption spectra are surprisingly accurate considering the strong approximations made and agree well with fullwave simulations. Our findings suggest experimental investigation of the designed woodpiles for chiral thermal emission. More fundamentally, they demonstrate the power of the concept of evanescent Floquet states to understand light propagation in finite slabs of artificially structured materials.
2 Methods
Evanescent Floquet states span the vector space of solutions of Maxwell’s equations within a finite slab of a periodic material [40]. While in theory, a countably infinite number of such modes exists at each frequency, a small number usually suffices in practice, making them an invaluable tool to not only predict but most importantly understand the physical origin of a scattering experiment. If a periodic optical material contains base materials with loss, such as plasmonic metals, even the fundamental Floquet modes do not fit into the standard band structure picture with realvalued wave vectors [3]. Instead, a complex band structure picture has to be adopted [40, 41]. Calculating these modes for general geometries remains challenging and is currently not possible with established software packages (based on for example finite differences or finite elements). We here present a semianalytical method to efficiently compute the complex band structure for the fundamental evanescent Floquet states of a hexagonal chiral woodpile. While our method cannot accurately predict scattering observables, it instead

1.
reveals the main physical mechanism of scattering even for frequencies close to the first Wood anomaly substantially above the homogenization regime, and

2.
allows for easily optimizing a desired behavior (broadband circular dichroism in this manuscript) within a large parameter space.
2.1 Single layer scattering
We start by calculating the Floquet states within each layer, which is homogeneous in z direction and whose local coordinate frame is defined in Fig. 1c. The procedure is based on a symmetry simplification of the wellknown lamellar grating equation [43]. At normal incidence, the electric (TE) or magnetic (TM) field points in ydirection, and is thus antisymmetric with respect to the mirror operation \(\sigma _y\) that maps \(y\,{\mapsto }\,{}y\). The relevant scalar (monochromatic) field \(F(\textbf{r})\) is the ycomponent of the electric (TE) or magnetic (TM) field. For the whole chiral slab, since we are interested in planewave excitation at normal incidence, we can characterize the field as odd with respect to the twofold rotation symmetry around the zaxis at the center of the unit cell \(C_{2z}\), that coincides with and results from multiple application of the \(C_{6_2}\)axis shown in cf. Fig. 1d. The scalar field \(F(\textbf{r})\) is, therefore, symmetric under \(\sigma _x\,{=}\,C_{2z}\sigma _y\) at the center of both the dielectric and the metal domain, shown as dashed lines in Fig. 1c.
Due to the homogeneity of the single layer in zdirection, the Floquet modes are planewavelike. The general monochromatic solution of Maxwell’s equations that satisfies the symmetry requirements in the domain \(\eta \,{=}\,1,2\), where \(1\,{\equiv }\,\text {dielectric}\) and \(2\,{\equiv }\,\text {metal}\), is hence (with \(\imath \,{:=}\,\sqrt{{}1}\))
Here, \(c_{\eta }\,{\in }\,{\mathbb {C}}\) is a complex coefficient, \(q\,{\in }\,{\mathbb {C}}\) the wave number in the propagation direction, \(x_{\eta }\) the center of the domain, and \(k_{\eta }\,{=}\,\pm \sqrt{\varepsilon _{\eta } k_0^2q^2}\) the lateral wave number given by the material dispersion relation, with \(k_0\,{:=}\,\omega /c_0\) the vacuum wave number (with \(\omega\) the angular frequency and \(c_0\) the speed of light). The solution at the interface between the two domains additionally requires the tangential components of the electric and the magnetic field to be continuous, which leads to
with the wave impedance \(Z_{\eta }\,{=}\,k_{\eta }/k_0\) (TE) and \(Z_{\eta }\,{=}\,k_{\eta }/(\varepsilon _{\eta }k_0)\) (TM) and the (generally complexvalued) optical phase \(\varphi _{\eta }\,{=}\,k_{\eta }d_{\eta }/2\). A countably infinite number of Floquet solutions is thus obtained from solving the root equation
This is evidently a transcendental problem as all \(\varphi _{\eta }\) and \(Z_{\eta }\) implicitly depend on q through the respective material dispersion relation. For the metaldielectric structure and the wavelength range under consideration, however, a good guess can be obtained analytically. For this, let us first discuss the function \(\lambda (q)\) in the complex plane. It inherits two branch points from the two root functions of \(k_{\eta }\) at \(q_{\eta }\,{=}\,\sqrt{\varepsilon _{\eta }} k_0\).
We now use the fact that these two roots are well separated (compared to \(k_0\)) in the midinfrared, where \(\Vert q_2\Vert \gg \Vert k_0\Vert\) for all metals. Due to the impedance mismatch between the two regions, the intensity of the loworder Floquet modes is either concentrated in the dielectric or in the metal region. In other words, solutions will be either found relatively close to \(q_1\) or close to \(q_2\) (and far away from the other branch point). Let us start with the first case, where the intensity is concentrated in the dielectric. Since the solution is far away from \(q_2\), the phase \(\varphi _2\) has a large, positive imaginary part \(\Im [\varphi _2]\,{\gg }\,1\). The root function thus simplifies to
For TE polarization, \(\Vert Z_2\Vert \,{\gg }\,\Vert Z_1\Vert\), while for TM polarization, \(\Vert Z_1\Vert \,{\gg }\,\Vert Z_2\Vert\), so that we obtain
If the fields laterally concentrate in the metal domain, the phase \(\varphi _1\) has a large, negative imaginary part \(\Im [\varphi _1]\,{\ll }\,{}1\), leading to
Since \(\Vert k_1\Vert \,{\gg }\,\Vert k_2\Vert\), and \(\Vert \varepsilon _1\Vert \,{\ll }\,\Vert \varepsilon _2\Vert\), we obtain \(\Vert Z_1\Vert \,{\gg }\,\Vert Z_2\Vert\) irrespective of polarization. The approximate sequence of roots for both polarizations is thus
In the remainder of this paper, we only consider the fundamental TE/TM air mode with \(\eta \,{=}\,\alpha \,{=}\,0\), which is sufficient to qualitatively explain the observed scattering physics for small metal fill fractions^{Footnote 1} and a lateral lattice constant below the first Wood anomaly (\(a\,{<}\,\lambda\)). The successful application of this crude approximation can be understood by considering the following two facts: First, higherorder dielectric modes have little intensity in the fundamental Bragg scattering order (the DC Fourier component in x). On the other hand, the metal modes exhibit a weak coupling to a vacuum plane wave due to the strong impedance mismatch. The exact roots of the fundamental mode can be found using a standard Newton procedure with (4) as an initial estimate. The fundamental and firstorder modes of a platinumair grating are shown in Fig. 2. Higherorder TE modes, where the Newton formalism does not converge with the analytical guess values, can be obtained using a global contourintegral method on a disk in the complex plane around the approximate higherorder root positions [51]. We illustrate the behavior of \(\lambda (q)\), including the branch cuts, the position of the roots, and the contour integral method in Fig. 3.
2.2 Chiral woodpile unit cell transfer matrix and complex band structure
We conveniently calculate the transfer matrix within one unit cell using the single layer transfer matrix \({\mathcal {T}}\) and the rotation matrix \({\mathcal {R}}\) between neighboring layers, as illustrated in Fig. 1d. The layer transfer matrix connects the parallel components of the fields \((E_x,E_y,H_x,H_y)\) at the top of the layer to those at its bottom in its native coordinate frame, shown in Fig. 1c, d.
We first introduce the impedance matrix Z that translates from the wave amplitudes of the Floquet states \(\textbf{f}\,{:=}\,(f_\mathrm{{TE}}^{(+)},f_\mathrm{{TM}}^{(+)},f_\mathrm{{TE}}^{()},f_\mathrm{{TM}}^{()})\), containing the TE/TM amplitudes of the downward (\(+\), with group velocity in positive z direction) and upward (–) waves \(f_\mathrm{{TE/TM}}^{(\pm )}\), to the parallel field components in the local coordinate frame.^{Footnote 2} It is thus given by
with the \(q_\mathrm{{TE}}\) and \(q_\mathrm{{TM}}\) the respective wave number of the fundamental Floquet mode within the air region \(q_1^{(0)}\), see (4). The corresponding propagation matrix for the Floquet states is
with \(p_\mathrm{{TE/TM}}^{(\pm )}\,{:=}\,\exp \{\pm \imath q_\mathrm{{TE/TM} }h\}\). Using the impedance and the propagation matrix defined above, we thus obtain for the layer transfer matrix
The rotation matrix between neighboring layers is
with the \(2\times 2\) identity matrix \(\mathbbm {1}\) and the Kronecker product \(\otimes\). The rotation angle is generally \(\theta \,{=}\,2\pi /N\) (\(N\,{\in }\,{\mathbb {N}}\)), with \(N\,{=}\,3\) for the hexagonal woodpile with full crystallographic symmetry discussed here. The corresponding unit cell transfer matrix is
The eigen decomposition
with a diagonal matrix \(\Lambda \,{=}\,\text {diag}(\lambda _\beta )\), trivially yields the complex Floquet band structure with the complex wave number
The corresponding eigenfield to the Floquet wave number \(\kappa _\beta\) is obtained through resubstituting the field components, given by the \(\beta\)th column of \(Z^{1}.V\), into (1).
2.3 Chiral woodpile scattering matrix
We here connect the Floquet states belonging to the complex band structure (10) to the scattering parameters reflectivity, transmissivity, and absorbance. We assume a slab of chiral woodpile on a substrate with isotropic (but generally frequency dependent) refractive index \(n_\mathrm{{s}}\), and a circularly polarized planewave excitation at normal incidence from the top (assumed to be vacuum).
We first introduce the impedance matrix \(Z_0(n)\) that translates from the circularly polarized planewave amplitudes \((f_{++},f_{+},f_{+},f_{})\) within a background with refractive index n to the fields at an the interface at \(z\,{=}\,0\) in the native coordinate frame, such that
The impedance matrix is thus given by
Naively, the transfer matrix through a finite slab of M unit cells of chiral woodpile is thus given by
Using (8) for the unit cell transfer matrix, this expression cannot compute scattering at a semiinfinite slab without substrate, becomes highly inefficient for thick slabs, and most importantly does not reveal the relation to the complex band structure. Using the eigen decomposition (9), however, the slab transfer matrix becomes
and for a semiinfinite slab
To extract physical meaning, we need to translate these transfer matrices into the corresponding scattering matrices. For this, we first sort the 4 Floquet solutions in (9) with the permutation \(\Pi\), such that the two waves propagating energy in positive z direction are stored first, and \(\Lambda _\Pi \,{:=}\,\Pi .\Lambda .\Pi ^\intercal\) and \(V_\Pi \,{:=}\,V.\Pi ^\intercal\).^{Footnote 3} We reexpress
in the sorted basis and can now subdivide all transfer matrices into \(2\,{\times }\,2\) subblocks that connect downward (\(+\)) and upward (−) moving amplitudes, respectively:
For the scattering matrix we use the most efficient convention [52] to relate the incoming to the outgoing amplitudes:
Note that we have included the \(2\,{\times }\,2\) phase matrices that transport the incoming amplitudes from the other end of a finite slab to the interface in question for numerical stability [53, 54]. Since we are interested in intensities only, these are the identity matrices in the sub and superstrate, and the respective subblocks of \(\Lambda _\Pi ^M\) within the chiral woodpile. We can thus express the scattering matrix in terms of the transfer matrix as
For the semiinfinite slab, we substitute \({\mathcal {T}}_\mathrm{{inf}}\) and \(P_1\,{=}\,\mathbbm {1}\) into (17a) to obtain the reflectivity matrix in the circular polarization basis \(R\,{=}\,\Vert S_{11}\Vert ^2\).^{Footnote 4} Energy conservation yields the absorptivity \(A_\mathrm{{inf}}^{(\sigma )}\,{=}\,1{}\sum _{\sigma '}R_{\sigma '\sigma }\) for left (\(\sigma \,{=}\,{}\)) and right (\(\sigma \,{=}\,{+}\)) circularly polarized incoming light (from the point of view of the receiver).
For thick finite slabs, the transfer matrix \({\mathcal {T}}_\mathrm{{slab}}\) in (12) becomes numerically illconditioned. This problem is wellknown [53], and even exists in the fundamental Bragg order here due to the possible strong evanescence of the fundamental Floquet modes (see Sect. 3). As a consequence, the application of (17) to the slab transfer matrix in (12) fails in practice. Instead, the scattering matrix \({\mathcal {S}}_\mathrm{{t}}\) at the top interface between the vacuum and the woodpile is well behaved and obtained by substituting \({\mathcal {T}}\,{=}\,V_\Pi ^{1}.Z_0(1)\), \(P_1\,{=}\,\mathbbm {1}\), and \(P_2\,{=}\,\Lambda _{\Pi ,}^{M}\) into (17). Similarly, the scattering matrix \({\mathcal {S}}_\mathrm{{b}}\) between the woodpile and the substrate is obtained by substituting \({\mathcal {T}}\,{=}\,Z_0^{1}(n_s).V_\Pi\), \(P_1\,{=}\,\Lambda _{\Pi ,+}^M\), and \(P_2\,{=}\,\mathbbm {1}\) into (17). The scattering matrix through an arbitrary finite slab of chiral woodpile is hence obtained in a numerically wellbehaved way through the application of the Redheffer star product [54, 57]
defined for \(2N\,{\times }\,2N\) matrices \(C\,{=}\,A\circledast B\) with \(N\,{\times }\,N\) subblocks as
The finite slab reflectivity matrix is then \(R\,{=}\,\Vert ({\mathcal {S}}_\mathrm{{slab}})_{11}\Vert ^2\), while the transmissivity matrix is \(T\,{=}\,n_s \Vert ({\mathcal {S}}_\mathrm{{slab}})_{21}\Vert ^2\). As for the semiinfinite woodpile, energy conservation yields the absorptivity. If the substrate is lossless, we obtain \(A_\mathrm{{slab}}^{(\sigma )}\,{=}\,1{}\sum _{\sigma '}\left( R_{\sigma '\sigma }{+}T_{\sigma '\sigma }\right)\) for left (\(\sigma \,{=}\,{}\)) and right (\(\sigma \,{=}\,{+}\)) circularly polarized incoming light. For a lossy substrate, we instead have \(A_\mathrm{{slab}}^{(\sigma )}\,{=}\,1{}\sum _{\sigma '}R_{\sigma '\sigma }\). We generally define the (spectral) circular dichroism, both for the slab and the semiinfinite chiral woodpile, as
and the spectrally averaged circular dichroism, averaged over the frequency range \(\Omega\) (from approximately 75THz to 100THz) as
2.4 fullwave simulations and materials
The semianalytical results have been compared to fullwave Maxwell simulations using COMSOL Multiphysics. The frequency domain simulations were performed on a lateral hexagonal unit cell, as shown in Fig. 1a, employing periodic boundary conditions and a tetrahedral finite element mesh with second order tetrahedral elements and a maximum edge length of a/10. We thus simulated a finite woodpile structure for optimized geometrical parameters and 15 unit cells slab thickness. An air domain and a substrate domain of 2c height were added above and below the woodpile, respectively. These vacuum and substrate domains were terminated by periodic ports, in which the system was excited through the vacuum port in the circularly polarized basis. The \({\mathcal {S}}\)parameters were extracted from the Rayleigh components of the fields at these ports.
The material parameters for the different metals were taken from the refractive index database [58]. We specifically investigated the plasmonic metals summarized in Table 1. As the substrate, we used fused silica glass with \(n_\mathrm{{s}}\approx 1.5\), silicon with \(n_\mathrm{{s}}\approx 3.5\), which were approximated as nondispersive and lossless materials, and the specific metal in question with strongly dispersive refractive index.
3 Results and discussion
3.1 Complex band structure of the chiral plasmonic woodpile
To understand the guiding principle behind light propagation in the chiral plasmonic woodpile structures, we start analyzing the evanescent Floquet states calculated by the algorithm introduced in Sect. 2.2. For a broad range of geometrical parameters, which will be discussed in more detail in Sect. 3.2, we find a polarization bandgap in the frequency range of interest \(\Omega\) between 3 and 4 \(\upmu\)m wavelength: On the one hand, the two fundamental Floquet modes either predominantly couple to left (LCPphilic) or right (RCPphilic) circular polarization. On the other hand, the LCPphilic mode exhibits a band structure with weak dispersion and a small imaginary part, while the RCPphilic mode gives rise to a large bandgap with a relatively large imaginary part and a real part close to the Brillouin zone boundary at \(\kappa \,{\approx }\,\pi /c\), as shown in Fig. 4.
While the polarization bandgap resembles that of a chiral high index photonic crystal, there are two distinct differences: First, the PC bandgap found here is above the frequency \(k_0\,{=}\,\pi /c\), while in a photonic crystal, it is below that frequency, simplifying topdown fabrication as the corresponding structures for a midIR target frequency will be bigger. Second, the RCPphilic mode is not exactly pinned to the Brillouin zone boundary at \(\kappa \,{=}\,\pi /c\), corresponding to a nonvanishing energy propagation and thus the expected finite energy loss of the strongly evanescent mode within the structure. Similarly, the LCPphilic mode has a small, but nonvanishing imaginary part of \(\kappa\), corresponding to a small evanescence or Beer–Lambertlike energy dissipation while propagating through the woodpile PC. As we will demonstrate in Sect. 3.3, the attenuation is underestimated by our approximate theory, as we do not consider the higherorder Floquet modes that concentrate their energy in the metal domain and thus give rise to an additional loss in the PC.
Regarding the circular polarization discrimination, the field polarization of the two Floquet states resides close to the north and south poles of the Poincaré sphere, respectively. We quantify the circular dichroism of the Floquet states with the circular dichroism index [12, 59]
that considers the relative difference in coupling to the two circularly polarized plane waves and the Floquet field and ranges from 0 (no difference) to \(\pm 1\) (maximum difference). The total incoupling from the vacuum is similarly quantified through the coupling index
that ranges from 0 (no incoupling) to 1 (maximum incoupling). The individual RCP and LCP couplings in these expressions are approximated by
where \(V_\Pi ^{(i)}\) (\(i\,{=}\,1,2\)) is the ith sorted eigenvector of the unit cell transfer matrix, that is the ith column of \(V_\Pi\), which contains the lateral electromagnetic fields of the corresponding Floquet state on the \(C_{6_2}\) axis at the top of the unit cell. As this definition considers both the electric and magnetic fields, it takes the impedance match into account.
We find that the circular dichroism index for the platinum structure corresponding to the band structure in Fig. 4 at wavelengths \(\lambda _0\,{=}\,(3,3.5,4)\,\upmu{{\text {m}}}\) is \({\mathcal {C}}\,{=}\,(0.84,0.83,0.81)\) for the LCPphilic mode and \({\mathcal {C}}\,{=}\,(0.98,0.94,0.89)\) for the RCPphilic mode. In other words, incoming LCP light couples almost exclusively into the blue, weakly attenuated propagating Floquet state, while RCP light couples predominantly into the red, strongly evanescent state in Fig. 4a, justifying the color assignment. Further, the coupling index is \(\beta \,{=}\,(0.99,0.99,0.98)\) into the LCPphilic mode, and \(\beta \,{=}\,(0.67,0.52,0.52)\) into the RCPphilic mode for \(\lambda _0\,{=}\,(3,3.5,4)\,\upmu{{\text {m}}}\). Even though the approximate theory is expected to overestimate the coupling, this suggests that almost all of the incoming LCP light is transmitted into the woodpile PC and absorbed through Beer attenuation within. On the other hand, a considerable amount of RCP light is expected to couple into the red, strongly evanescent Floquet mode. Since this mode, however, propagates almost no energy into the structure within the bandgap region, RCP light is mainly reflected with little absorption.
3.2 Optimizing broadband circular dichroism
While the qualitative chirooptical behavior of the chiral woodpile PC is well understood within the band structure picture discussed in Sect. 3.1, we here perform a more quantitative analysis calculating the scattering matrix and the circular dichroism as outlined in Sect. 2.3.
Next to revealing the main physical scattering mechanism, the semianalytical Floquet mode algorithm comes at a very low numerical cost. The scattering parameters for one wavelength are calculated within 2 ms (in a nonoptimized Python implementation), compared to 5 min for the fullwave simulations (see Sect. 2.4 for details). This efficiency enables a bruteforce scan over geometrical parameters for different materials to extract the spectrally averaged circular dichroism \(\langle \textrm{CD}\rangle\), defined in (20). We find, that a strong CD can be generally observed for lattice constants below, but on the order of 3 \(\upmu{{\text {m}}}\), and have therefore fixed the single layer pitch to \(d\,{=}\,{2.4}\,\upmu {{\text {m}}}\) (\(a\,{\approx }\,{2.77}\,\upmu {{\text {m}}}\)) for the remainder of this manuscript.
For the four metals listed in Table 1, we have calculated the average CD of the semiinfinite metalair PC with varying thickness of the inplane air region \(d_1\,{\in }\,[1,2]\,\upmu {{\text {m}}}\) and the layer height \(h\,{\in }\,[0.3,1.5]\,\upmu {{\text {m}}}\). Note that the metal fill fraction and the chiral pitch are hence varied according to \(\phi \,{=}\,1{}d_1/d\) and \(c\,{=}\,3h\). The results illustrated in Fig. 5 reveal a broad region within the parameter space, where a substantial CD can be observed, largely independent of the chosen metal. This region between 600 and 800 nm layer heights, where the spectrally averaged CD is above 0.5, spans almost all \(d_1\), with a slight increase towards larger thicknesses. While aluminum and tungsten yield the strongest CD for the semiinfinite slab (cf. Table 1), their more strongly perfect plasmonic nature (\(\Im \{n\}\,{\gg }\,\Re \{n\}\)) gives rise to very little field penetration into the metal region. This in turn makes them less efficient in more realistic finite slabs, since the attenuation of the LCPphilic mode (Sect. 3.1) and thus LCP absorption is much weaker. As discussed in Sect. 3.3, attenuation is, however, underestimated by our theoretical approximation. The chiral PC is expected to yield a substantial spectrally averaged CD, (20), of 70% for all metals, insensitive to fabrication imperfections. A specific metal alongside geometrical parameters within a broad region may hence be chosen depending on the limitations of a specific fabrication routine.
In summary, the CD mainly depends on \(d_1\) and h (and not explicitly on a) and is strongest within a specific region in Fig. 5, which is independent of the metal in question. To better understand this behavior, we approximate the chiral woodpile even further by replacing it by a plasmonic version of a semicontinuous Bouligand structure, as found in liquid crystals and biological systems [60]. Since we expect the electromagnetic field to rotate with the smaller 60\(^\circ\) rotation of the \(C_6\) screw rotation, and not the 120\(^\circ\) of the \(C_3\) rotation used in Sect. 2.2 and illustrated in Fig. 1, we build the Bouligand structure such that it rotates in the opposite direction and has a pitch of 2c. The anisotropic lateral (xy) permittivity matrix depending on \(\varepsilon _\mathrm{{TE/TM}}\,{:=}\,q_\mathrm{{TE/TM}}/k_0\) at height z is in the Bouligand picture expressed by:
The monochromatic Maxwell equations can be solved with the ansatz
for the lateral electric field (with \(E_z\,{=}\,0\)). With the approximate solutions for the Floquet wave numbers (4), this procedure yields the following 2D quadratic eigenproblem in \(\kappa\):
with \(G_1\,{:=}\,\pi /d_1\). One can immediately see, that this approximate Bouligand equation only depends on the chiral pitch c (or layer height h) and thickness of the air region \(d_1\).
Let us first discuss the solution to (25) for \(k_0 \,{=}\,G\), for which the characteristic equation yields eigenpairs \((\kappa ,\textbf{E}_0)\) for the downward propagating waves
The R branch corresponds to the lower end of the bandgap in Fig. 4 with \(\kappa _\mathrm{{R}}\,{\approx }\,0\).^{Footnote 5} The mode is, however, linearly polarized in y direction. On the other hand, the LCPphilic solution is indeed left elliptically polarized, if \(\kappa _\mathrm{{L}}\) is real, that is if \(h\,{<}\,\frac{2}{3}d_1\). At the upper end of the bandgap at \(k_0\,{=}\,\sqrt{G^2+G_1^2}\) we obtain a similar solution
The R branch is now linearly polarized in x direction. While the L branch is always left elliptically polarized, it approaches circular polarization if \(h\,{\ll }\,\frac{2}{3}d_1\). These two branches are connected by the solution within the center of the gap at \(k_0\,{=}\,\sqrt{G^2+G_1^2/2}\):
Evidently, \(\kappa _\) is purely imaginary, as expected in the center of the bandgap. It connects the \(\kappa _\mathrm{{R}}\) solutions and is linearly polarized with polarization direction between x and y. On the other hand, \(\kappa _+\) belongs to the \(\kappa _\mathrm{{L}}\) branch and is left elliptically polarized, quickly approaching circular polarization if \(h\,{\ll }\,\frac{2}{3}d_1\).
The Bouligand model thus identifies three main contributors that limit the region of high average spectral CD. First, the \(\kappa _\mathrm{{R}}\) bandgap needs to reside within the spectral range of interest. To keep the lower band edge outside the spectral region of interest, we require \(G\,{<}\,2\pi /\lambda _\mathrm{{l}}\), with \(\lambda _\mathrm{{l}}\,{=}\,{4}\, \upmu {\text {m}}\) in our case. This yields \(h\,{>}\,\lambda _\mathrm{{l}}/6\,{\approx }\,{0.67}\, \upmu {\text {m}}\), shown as brown line in Fig. 5, which explains the horizontal border at the bottom of the high CD domain. Similarly, keeping the upper bandgap edge outside the spectral region of interest requires
(where the radicand is positive), shown as a red line in Fig. 5. Above this line, the red region in the heatmap indicates a sign change in the average CD. This change is caused by a dichroic color switch [61], meaning that there is an additional bandgap of opposite optical chirality at higher frequencies, which also exists in chiral woodpile photonic crystals [38] and moves into the spectral region of interest.
Finally, the polarization of the \(\kappa _\mathrm{{L}}\) mode needs to resemble LCP polarization, i.e. it needs to be as close as possible to the pole of the Poincaré sphere, to yield a strong CD in the bandgap. This implies that we need to be as far as possible under the white \(h\,{=}\,\frac{2}{3}d_1\) line in Fig. 5, explaining the less sharp positively sloped upper termination of the high CD region. While the Bouligand model can thus explain the general shape of the high CD region, it predicts the red bandgap mode to be linearly polarized in contrast to the hexagonal chiral woodpile bandgap modes, which are clearly right circularly polarized as demonstrated by the CD index in Sec. 3.1.
3.3 Comparison to fullwave simulations
Considering the crude approximations made, the theory predicts the full spectral CD of all metals well. To demonstrate this, we have calculated the RCP and LCP absorption through fullwave simulations as described in Sec. 2.4. The results for an ironair woodpile PC on a glass substrate with \(d\,{=}\,{2.4}\, \upmu {\text {m}}\), \(\varphi \,{=}\,0.2\), and \(h_1\,{=}\,{0.7}\, \upmu {\text {m}}\), close to the optimal parameters of the semiinfinite slab, are shown in Fig. 6. Clearly, the position of the bandgap is predicted well, although the fullwave results seem to be slightly blueshifted compared to the theory. Generally, the theory underestimates the absorption for both polarizations within the bandgap region. This behavior is expected, as we do not consider the single layer higher order fields that reside mainly in the metal domain. These lead to stronger absorption both in the evanescent Floquet mode (red spectrum), and the propagating mode (blue spectrum). An additional indication that absorption and hence attenuation in the propagating Floquet mode is increased in the simulations is the absence of the Fabry–Pérot interference pattern that is clearly visible in the theoretical spectrum. At high wavelengths above \({4}\, \upmu {\text {m}}\) on the other hand, where iron begins to act more and more like a perfect electrical conductor, the inclusion of the higher order metallic modes is expected to lead to an underestimated impedance mismatch between the incoming vacuum field and the Floquet states within. The theory, therefore, underestimates the reflection and overestimates the absorption for both polarizations alike.
While the theory cannot accurately predict the spectra, we have thus demonstrated that all qualitative physical predictions made in the preceding chapters are accurate and can be used to tailor a chiral woodpile PC to a specific application and fabrication procedure. These general principles can also help to further finetune the geometry to improve the nonapproximated CD. For example, the position of the bandgap in Fig. 6 seems to be too far to the left. This suggests increasing the layer height h slightly, which is predicted to redshift the right band edge towards 4 \(\upmu\)m, while only weakly affecting the left edge at 3 \(\upmu\)m. We demonstrate this effect by simulating the same structure with an increased \(h_2\,{=}\,{0.72}\, \upmu {\text {m}}\), shown as dotted line in Fig. 6.
4 Conclusion
In conclusion, we have identified the underlying physical principles of broadband circular dichroism in chiral plasmonic woodpile structures employing an approximate evanescent Floquet mode picture. Focusing on the transparency window of the atmosphere between 3 and 4\(\upmu\)m wavelength in the midinfrared frequency region, we have found a broad circular polarization bandgap that exists within a large region of geometrical parameters and for a number of different metals used. Employing a semicontinuous Bouligand model, we extracted general design principles that predict the approximate size and shape of the region in the geometrical parameter space, where large broadband circular dichroism is expected.
On the one hand, our findings demonstrate that evanescent Floquet modes and the associated complex bandstructure form an invaluable tool to understand scattering at and wave propagation within slablike plasmonic crystals, which combine interferencedominated physics known from photonic crystals and materialdispersion induced effects known from classical metamaterials. On the other hand, we provide a pathway to design a broadband, highly efficient circularly polarized thermal source in the midinfrared region by application of Kirchhoff’s law. A large broadband CD can be engineered within a predictable, massive region in the geometrical parameter space. This region encompasses a variable aspect ratio of the metal bars, which ranges from a ratio of approximately 1 : 2 to 2 : 1. The typical lattice constant is smaller, but comparable to the wavelength of the light, approximately twice as big as in highindex dielectric structures, making topdown fabrication more feasible for the envisioned midinfrared window between 3 and 4\(\upmu\)m wavelength.
Indeed, a number of fabrication routines have been reported to yield woodpile structures for geometrical parameters within the predicted highCD region, ranging from layerbylayer manufacturing [62, 63] to twophoton lithography techniques [64]. Recent advances in twophoton lithography make it possible to directly produce metallic structures [65], while all other methods can produce an inverse mold on a conducting substrate (for example indium tin oxide). In both cases, an electrodeposition routine [66] can replicate the woodpile PC in a metal of choice. Standard direct laser writing on the other hand typically produces a polymeric woodpile structure, for which an electroless plating routine can be employed [67, 68].
Notes
The model is expected to produce good results if the thickness of the metal domain is above the effective penetration depth of the fields on the order of 100 nm as seen in Fig. 2, but such that \(\phi \,{\ll }\,1\).
Note that this impedance matrix formally resembles that of a homogenized view, but the layer propagation constants and fields are exact so that the approximation lies in cutting the Fourier series in its fundamental order when matching the fields between neighboring layers, not within.
Since energy is lost within the chiral woodpile, this simply requires sorting by \(\Vert \lambda _\beta \Vert\) in ascending order.
This simple expression is a consequence of all waves in the circularly polarized basis to be normalized such that their energy flow in propagation direction is twice the background index times the squared absolute value of the amplitude.
Note that the boundary of the Brillouin zone is backreflected to the \(\Gamma\) point at the center in the considered nontrivial unit cell with pitch 2c.
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Acknowledgements
We would like to thank JeanJacques Greffet and Cedric Schumacher for useful discussions.
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Open access funding provided by University of Fribourg Ullrich Steiner acknowledges funding from the Swiss National Science Foundation through Project grant 188647.
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Abdennadher, B., Iseli, R., Steiner, U. et al. Broadband circular dichroism in chiral plasmonic woodpiles. Appl. Phys. A 129, 229 (2023). https://doi.org/10.1007/s00339023064819
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DOI: https://doi.org/10.1007/s00339023064819
Keywords
 Complex bandstructure
 Circular dichroism
 Plasmonic crystal
 Chiral woodpile
 Thermal emission