## Abstract

Metal-helix based metamaterials have been introduced as compact and broadband circular
polarizers. However, the end of the metal wire together with the helix center defines an axis
in space, which unavoidably breaks the rotational symmetry at the metamaterial surface. This
introduces linear birefringence. Symmetry can be recovered by considering an integer number,
e.g. *N* = 4, of intertwined helices arranged to a square array. We show that
the operation principles are fundamentally different though. Metamaterial circular polarizers
based on *N* = 4 helices, unlike single helices, inherently require absorption
of the constituent metal. Otherwise, the combination of a four-fold rotational axis and
time-inversion symmetry strictly forbids circular-polarizer action. Our symmetry analysis is
confirmed by extensive numerical calculations comparing results for perfect electric conductors
with those for a free-electron Drude metal with finite damping.

©2012 Optical Society of America

## 1. Introduction

Gold-helix metamaterials have been introduced as compact and broadband circular polarizers [1–3]. They can be seen as the circular counterpart of the good old wire-grid polarizer for linear polarization of light. Simple square arrays of circular helices with two and more pitches along the helix axis provide an operation bandwidth of nearly one octave [1] (for light propagating along the helix axis). Within this bandwidth, one circular polarization is transmitted, the other one is reflected. Metal absorption is not required and is actually rather unwanted. Several variations of this idea have meanwhile been published aiming at further improving the circular-polarizer performance. For example, three-dimensional bi-chiral arrangements [4,5] promise a more isotropic response, however, they largely deteriorate both the bandwidth and the extinction ratio [5]. Tapered helices increase the bandwidth to about 1.5 octaves, increase the extinction, and reduce polarization conversion [6]. Simplified layer-by-layer chiral designs [7–9] cannot quite match the performance of helix-based metamaterials, but these structures are closer to practical applications and allow for much larger sample footprints due to an inherently simpler fabrication process [9].

When looking at a circular helix along the helix axis (see Fig. 1(a) ), the end of the wire forming the helix together with the central axis of the helix form a direction in space, which breaks the wanted full rotational symmetry of the chiral metamaterial [2]. In transmission, this fact leads to a small but unwanted degree of circular polarization conversion, e.g., incident left-handed circular polarization (LCP) is partially converted to emerging right-handed circular polarization (RCP). In reflection, this conversion effect is dramatic [2]. For a bulk metal, a bulk dielectric, or for an ideal isotropic chiral effective material under normal incidence of light onto a sample surface, e.g., LCP would be reflected as pure RCP. In sharp contrast, for metal-helix metamaterials, incident LCP is reflected as LCP, i.e., one gets nearly 100% of polarization conversion in reflection [2].

To reduce or eliminate these unwanted polarization conversion effects, multiple (integer
*N*) intertwined helices have previously been proposed [10,11]. For *N* = 2,
they can be seen as the analog to the DNA double strands in biology. At first sight, one might
be tempted to look at an *N*-helix metamaterial as a small modification of a
metamaterial composed of single helices. In this work, we show that the operation principles are
fundamentally different due to different symmetry. As a result, the optimization of the
circular-polarizer action is very different, too.

## 2. Symmetry based analysis

Figure 1 illustrates unit cells of simple square arrays
of single helices and multiple helices for *N* = 4. In the *N* = 4
case in (b), the symmetry of a single building block (“meta-atom”) and that of the array are
identical. Both exhibit a four-fold rotational symmetry. *N* = 1, 2, 3, … do not
fulfill this condition, *N* = 8, 12, 16,… would. However, the difficulty of
fabrication obviously rapidly increases with increasing *N*. Thus, we only
explicitly consider *N* = 4 in this letter. It is known that the operation
bandwidth depends on the number of helix pitches along the helix axis [1,2]. We start our discussion with single
pitches, where the resonances are simpler to understand [1]. At the end of this paper, we also consider several axial pitches for
*N* = 4 helices. The dependence of the optical properties of single helices on
the number of pitches resembles the expectation from end-fire helical antenna theory [12], has previously been published [1,2], and shall not be repeated here.

For single helices, we have previously shown [1] that
the circular-polarizer effect survives if the metal is described as a perfect electric conductor
(PEC). For a PEC, absorption is strictly zero by definition. Thus, any difference in the
metamaterial transmittance for incident LCP and RCP must be due to a difference in reflectance
at the interfaces. For the case of *N* = 4 helices we now show that, due to
symmetry, the intensity reflectance and transmittance coefficients for LCP and RCP are strictly
identical.

Importantly, we shall not use the effective-medium approximation, the validity of which is
somewhat questionable under our conditions, but rather consider the sample as a “black box” and
only exploit its symmetries. Consider a plane wave of light impinging along the axis of the
helices, i.e., along the *z*-axis. Thus, the electric field has components in the
*xy*-plane. The reflected field vector _{${\overrightarrow{E}}_{\text{r}}$}can be connected to the incident field vector
_{${\overrightarrow{E}}_{\text{i}}$} by the field-reflection 2x2 matrix
$\overleftrightarrow{r}$ with complex-valued coefficients given by

_{${\overleftrightarrow{M}}_{90}.$}This is equivalent to invariance under the coordinate transformation

_{$\overleftrightarrow{r}={\overleftrightarrow{M}}_{90}\text{}\overleftrightarrow{r}\text{}{\overleftrightarrow{M}}_{90}^{-1}.$}This condition leads to

_{${r}_{xx}={r}_{yy}$}and

_{${r}_{yx}=-{r}_{xy}.$}Thus we have

_{${t}_{\text{LCPLCP}}$}and

_{${t}_{\text{RCPRCP}}$}and zero off-diagonal elements.

Next, we exploit time-inversion symmetry. Consider an incident field vector in the circular basis, _{${\overrightarrow{E}}_{\text{i,circ}},$} with components _{${E}_{\text{i,LCP}},$} and _{${E}_{\text{i,RCP}}.$} As the conversions are zero, we can, for example, consider solely _{${E}_{\text{i,LCP}}.$} This leads to the transmitted and reflected field vector components

_{$t\to -t,$}but also the wave vector

_{$\overrightarrow{k}\to -\overrightarrow{k}$}and any relative phase between the two vector components have to be inverted. Hence, the field vectors turn into their complex conjugates, see, e.g [13–15]. We now consider

_{${E}_{\text{t,LCP}}^{*}$}and

_{${E}_{\text{r,RCP}}^{*}$}as the incident field vector components. In general, the structure must not necessarily be symmetric with respect to inverting the propagation direction. For the incident field ${E}_{\text{t,LCP}}^{*},$ that is propagating in the opposite direction we therefore introduce the reflection and transmission matrices

_{$\overleftrightarrow{r}{\text{'}}_{\text{circ}}$}and

_{$\overleftrightarrow{t}{\text{'}}_{\text{circ}}.$}Both matrices will again have zero off-diagonal elements due to four-fold rotational symmetry. Due to time-inversion symmetry [15], the following two equations must be satisfied

Altogether, lossless *N* = 4 helices exhibit

- 1. four-fold rotational symmetry along the helix axis and
- 2. time-inversion symmetry (no losses and no static magnetic field).

*N*= 4 helix structure is chiral – but its optical properties show hardly any traces of that.

*In particular, the structure just cannot act as a circular polarizer.*

However, finite losses would break time-inversion symmetry. In this case, different transmissions may result from finite propagation losses that can be different for LCP and RCP respectively due to the chiral nature of the helices.

## 3. Numerical calculations

To test our symmetry considerations, we compare them to numerical calculations using the commercial software package *CST Microwave Studio* based on a finite-integration technique. The structure parameters for *N* = 1 and *N* = 4 are illustrated in Fig. 1(a) and 1(b) respectively. We choose *a* = 1 µm. We have performed calculations with and without the glass substrate, which does break the symmetry with respect to inverting the propagation direction [6]. The differences turn out to be very small though. Thus, for clarity, we only depict results without glass substrate here. For the frequencies depicted in Figs. 2
and 3
, diffraction into free space cannot occur (the high-frequency end of about 300 THz corresponds to a free-space wavelength of 1 µm = *a*).

We only show results for left-handed helices. For right-handed helices, LCP just needs to be exchanged by RCP and vice versa (by symmetry).

We start our discussion with the conceptually simple case of metal helices made of a PEC, for which losses are strictly zero. It becomes obvious from Fig. 2(a) that the single-helix metamaterial does act as a circular polarizer, whereas the *N* = 4 helix metamaterial in Fig. 2(b) shows strictly zero difference in intensity reflectance and transmittance between LCP and RCP at any frequency. This highlights the fundamental difference between the *N* = 1 and the *N* = 4 case, in agreement with our symmetry considerations in the preceding section. Energy conservation for a PEC means that the sum of all red curves in Fig. 2(a) at a given frequency is one (same for Fig. 2(b)). The same holds true for the blue curves. Our numerical calculations obey energy conservation to better than 1%. Within the PEC limit, due to the scalability of the Maxwell equations, the operation frequency can simply be scaled to any desired frequency.

However, for small structures and high operation frequencies, the PEC assumption becomes unrealistic and the metal rather has to be described by the Drude free-electron model, which does lead to finite losses. Corresponding calculations for gold with plasma frequency _{${\omega}_{\text{PL}}=1.37\times {10}^{16}\text{rad}/\text{s}$} and collision frequency _{${\omega}_{\text{col}}=1.2\times {10}^{14}\text{rad}/\text{s}$} are shown in Fig. 2(c) and (d), where again *a* = 1 µm. When going from PEC to gold, the performance of the single-helix metamaterial in (c) deteriorates with respect to (a). In sharp contrast, the PEC *N* = 4 helix metamaterial in Fig. 2(b) shows no circular-polarizer action at all, while at least some difference between LCP and RCP intensity transmittance is found in (d). This highlights once again the fundamental difference in operation principles between the *N* = 1 and the *N* = 4 case. In agreement with our symmetry considerations in the preceding section, all polarization conversions (dashed curves) are strictly zero for the *N* = 4 case, with and without losses.

So far, we have only discussed metamaterials composed of single helix pitches in this paper. The *N* = 1 case is known [1] to exhibit significantly larger bandwidth for two or more pitches along the helix axis, which is identical to the propagation direction of light. Figure 3 shows that the behavior is similar for the *N* = 4 helix metamaterial. When going from Fig. 3(a) to 3(d), the extinction depth increases with increasing number of pitches. This is simply due to the fact that one circular polarization (LCP) experiences much larger resonant absorption losses than the other one (RCP). Thus, with increasing number of pitches, hence increasing sample thickness, the difference in optical density between LCP and RCP increases. At the same time, the operation bandwidth (see gray areas) increases from one to six pitches. Intuitively, the bandwidth larger than a factor of three (i.e., 1.5 octaves) stems from the strong interaction among the different helix pitches [1], somewhat similar to the interaction of the primary and the secondary coil in a transformer [2].At the same time, the operation bandwidth (see gray areas) increases from one to six pitches. Intuitively, the bandwidth larger than a factor of three (i.e., 1.5 octaves) stems from the strong interaction among the different helix pitches [1], somewhat similar to the interaction of the primary and the secondary coil in a transformer [2].

Finally, we briefly mention that our above symmetry reasoning also applies to the metamaterial structure illustrated in Fig. 1(c). Here, four-fold rotational symmetry is recovered by changing the rotation of the adjacent *N* = 1 helices in one larger unit cell. Our corresponding numerical calculations (not depicted) exhibit zero circular polarization conversion for transmittance and reflectance and strictly no circular-polarizer action for any frequency within the PEC limit, just like for the *N* = 4 helices in Fig. 2(b).

## 4. Conclusions

In conclusion, due to four-fold rotational symmetry, the optical properties of metamaterials based on square arrays of *N* = 4 helices are fundamentally different from those of metamaterials based on *N* = 1 helices. For the *N* = 4 case, one does completely eliminate circular polarization conversion for both reflection and transmission, whereas polarization conversion is inherently unavoidable for *N* = 1. Unfortunately, without metal losses, one also completely eliminates the desired circular-polarizer effect for *N* = 4. The circular-polarizer effect can be recovered for finite metal losses. This implies that the structure is no longer easily scalable to any operation frequency and that certain inherent losses must be accepted. On the positive side, circular polarization conversion is strictly zero for *N* = 4 in reflection and transmission even in the presence of losses.

Normal direct laser writing (DLW) has allowed for realizing single-helix metamaterials at operation wavelengths of a few micrometers, but fabrication of *N* = 4 helices in that range presently appears out of reach with regular DLW. However, recent progress regarding stimulated-emission-depletion (STED) DLW [18] raises hopes that even metallic *N* = 4 helix metamaterials may become experimental reality in the not-too-distant future.

## Acknowledgments

We thank Ralph von Baltz (KIT) for discussions. We acknowledge support by the Deutsche Forschungsgemeinschaft (DFG), the State of Baden-Württemberg, and the Karlsruhe Institute of Technology (KIT) through the DFG-Center for Functional Nanostructures (CFN) within subprojects A1.4 and A1.5. The project METAMAT is supported by the Bundesministerium für Bildung und Forschung (BMBF). The PhD education of J. K. is embedded in the Karlsruhe School of Optics & Photonics (KSOP).

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