## Abstract

In this study, we present a comprehensive analysis to examine the origin of circular polarization stop bands in a dielectric helix structure. We show that band gaps in a helix structure may result from Bragg resonance or non-Bragg mechanism. The two types of gaps exhibit distinct optical properties and display an opposite dependence with respect to structural periodicity. The interplay of gaps not only gives rise to various operation scenarios, but results in pronounced modifications to dispersion characteristics that lead to abnormal propagation properties of circularly polarized waves. Our findings reveal versatile behaviors of circularly polarized light interacted with a three-dimensional helix medium, which can be of great importance for the design and implementation of circular polarization-dependent devices and applications.

© 2017 Optical Society of America

## 1. Introduction

Chiral materials exhibit a variety of optical properties and have attracted considerable attention in many applications. Chiral structures can be implemented by multiple stacking of planar structures [1,2], gyroid [3,4], and helices [5–7]. The chiral responses are of great interest in realizing artificial control of reflection/transmission of circular-polarized light and optical activity. By virtue of these unique characteristics, giant optical rotation [8], circular dichroism [9], and polarization stop bands [10] have been demonstrated based on chiral photonic crystals.

Helix is an important structure to study chiral responses as the helical geometry can well resemble the feature of circularly polarized (CP) light. Research momentum on dielectric helix structures has been sustained for decades, which is partially associated with the development of nanofabrication technologies. Many relevant studies on dielectric helix structures are discussed based on the fabrication platform of glancing angle deposition (GLAD) [11–17]. Corresponding analytical treatments have also been extensively studied [7,18–22]. Direct laser writing (DLW) is another common platform to fabricate 3D helix structures, where 3D structures can be written on the photoresist via precision movement of laser light by virtue of local polymerization induced by multi-photon absorption [10, 23–25]. Holographic lithography [26, 27] and focused ion/electron beam induced deposition [28,29] are also reported to fabricate 3D helix structures. In spite of very different geometries of helix structures based on various fabrication platforms, these studies are predominately focused on presenting the existence of gaps. Even for numerical studies of dielectric helix structures [5,30–32], analyses are emphasized on the presence of gaps with different structural parameters along with chirality analysis. In review of these numerical as well as experimental characterizations, efforts are mainly devoted to demonstrate the existence of band gaps, without addressing the underlying mechanisms.

In this study, we present the investigation on the origin of band gaps in a 3D dielectric helix structure. We show that band gaps may arise from Bragg resonance or non-Bragg mechanism as illustrated in Fig. 1. While the former is owing to the Bragg resonance associated with longitudinal period *p*, the latter is attributed to strong coupling between incident light and the local resonance of the structure governed by lateral period *a*. Previously, reflection or scattering characteristics due to non-Bragg mechanisms have been studied in other structures, including periodic waveguides, composite transmission lines, colloidal films, and phononic crystals [33–37]. Owing to different mechanisms, Bragg gaps and non-Bragg gaps normally exhibit distinct optical properties and different degrees of dependence with respect to structural parameters. We show that the 3D helix configuration has the interesting properties to be operated in various scenarios by virtue of the interplay between the two types of gaps, giving rise to polarization stop bands with different features and abnormal propagation properties of circularly polarized waves. An analytical model based on Bruggeman formalism and numerical simulations are presented to characterize the optical properties. Our findings reveal versatile behaviors of circularly polarized light interacted with a 3D helix medium that may also have implications for the study of other 3D chiral periodic structures.

## 2. Numerical calculation setup

The helix structures under study consist of right-handed (RH) helices arranged in a square lattice. The geometrical parameters are defined by the helix radius *R*, wire radius *r*, pitch length *p*, helix spacing *a*, and pitch number *N*. The geometrical parameters and the symmetry points within the first Brillouin zone in k-space are adopted from [32]. In this study, we focus our discussion on the first gap along the Γ*Z* direction, i.e., along the axis of the helix (z direction). The band structures are calculated by the plane wave expansion method implemented in Rsoft BandSOLVE software package and reflection spectra are calculated based on finite-difference time-domain (FDTD) method using Lumerical FDTD Solutions. In reflectance spectrum calculation, periodic boundary conditions are imposed in x and y directions and PML boundary conditions are imposed in z direction. The refractive index of the dielectric is 2.5. Unless otherwise specified, the parameters of the helix are *r* = 0.05*μ*m, *R* = 0.15*μ*m, *p* = 0.5*μ*m, and *N* = 10.

To evaluate the chirality of modes, normalized CP coupling coefficients, *η _{RH}* and

*η*, are calculated by the overlap integrals between the magnetic field

_{LH}*H⃗*(

*x*,

*y*,

*z*) of Bloch modes and a perfect CP wave.

*η*and

_{RH}*η*are defined as [9]

_{LH}*q⃗*representing the wave vector of the incident CP wave. The relative difference between coupling coefficients is used to quantitatively evaluate the degree of circular polarization. For simplicity, the notations of RH and LH circularly polarized light (mode) are denoted hereafter as RH and LH light (mode), respectively.

## 3. Results and discussions

#### 3.1. Evolution of band gaps in a dielectric helix structure

We begin our discussion with an illustration to exemplify the evolution of band gaps in a dielectric helix structure. We focus on the optical properties of RH light to elucidate the phenomena. Figure 2(a) depicts the reflection of RH light when the helix lateral period *a* varies from 0.3*μ*m to 0.7*μ*m. The color gradient represents the reflection level, by which the gap regions are denoted in red. At *a* = 0.3*μ*m, the lowest RH gap appears around 460THz, and it slightly shifts to high frequencies with increased lateral period. The reflection spectrum and the band structure at *a* = 0.35*μ*m are shown in Fig. 2(b). The RH gap is shaded in blue, where the corresponding dispersion curves around the gap can be identified in the band structure. Note that the band-edges are aligned in k-space and situated at the edge of the second Brillouin zone, that is, *k* = *λ/p* = 1. It means that the resonance occurs when the pitch length of the helix matches that of the circularly polarized light. For comparison, the reflection spectrum for the structure with one pitch (*N* = 1) is plotted as the dashed line. We can see that the resonance is barely discernible with a single pitch. These properties feature the characteristics of a Bragg band gap, the appearance of which is attributed to multiple scattering and interference associated with longitudinal periodicity.

As the helix lateral period *a* continues increasing, there is another high reflection band approaching from the top. The reflection spectrum and band structure at *a* = 0.44*μ*m are shown in Fig. 2(c). We can see that a high reflection region consists of two bands in close proximity to each other, where the dispersion curve in the middle represents the junction of bands. When helices are separated further apart, the merged region splits into two bands again. The reflection spectrum and band structure at *a* = 0.5*μ*m are shown in Fig. 2(d). Compared to the band structure in Fig. 2(b), the dispersion of the lowest RH gap exhibits different optical characteristics. The upper edge of the RH gap depicted by the yellow band is not aligned at the boundary of the Brillouin zone. Furthermore, the dashed line, representing the reflectance for *N* = 1, shows that total reflection can be obtained with a single pitch. These features are distinct from those of Bragg resonance. We use non-Bragg gap to refer to it in the following discussion.

From the above observations, we can see that the gap of the structure at first exhibits Bragg resonance behavior for smaller values of lateral period. By adjusting the helix spacing, another non-Bragg gap approaching from high frequency may cross the Bragg gap and then take over to be the lowest gap appeared in the spectrum. The different characters of the two bands can be clearly distinguished by their optical properties. Therefore, the helix structure may exhibit a photonic gap operated in different resonance regimes subject to geometrical variations.

### 3.1.1. Bragg gap regime

To characterize optical properties of the helix structure, we start with an approach previously used to formulate light propagation in helicoidal bianisotropic media [21, 38]. This general approach is described in the Appendix and has been applied to interpret the optical properties for a variety of chiral media, such as biaxial Reusch piles, cholesteric liquid crystals, etc. For these media, it is normally difficult to derive exact solutions from Maxwell’s equations. Therefore, a common approach is to conceptualize the media by a composite system, the properties of which can be described by effective constitutive parameters and the Bruggeman formalism is used to homogenize the composite [21]. In our calculation, the helical inclusion is assumed lossless with an index refraction of 2.5. The helical inclusion has a spherical cross-section and a slenderness aspect ratio of 15. The volume fraction based on this ellipsoid-based model will be underestimated compared to our structure. Therefore, we use a modified volume fraction *f′* = *V _{r}*

*f*to account for the geometry variation, where

*V*is the volume ratio of a cylinder to an ellipsoid.

_{r}To verify the validity of the model to the present helix structure, we solve the dyadic equation and compute the predicted central frequency of the Bragg gap ${\lambda}_{0}^{\mathit{Br}}$ based on equation (9) in the Appendix using the geometrical parameters of the helix structure in Fig. 2(a). The predicted central frequencies for different values of *a* are traced by the white curve. We can see that the curve follows the high reflection band for the region with *a* < 0.44*μ*m. It means that the homogenization approach can be employed to model optical properties of the present structure fairly well in the Bragg regime. The same calculations are also carried out to characterize the Bragg regime with different geometrical parameters. Figure 3(a) illustrates forbidden gaps with respect to lateral period *a*, where yellow, green, and blue bands are the gap regions for *p* = 0.6, 0.8, and 1*μ*m, respectively. By inspecting the evolution of bands, we can divide the operation regime into two. For smaller values of *a* or, equivalently, a more densely-packed helix array, the band belongs to Bragg gap, showing a modest blue-shift with *a*. For larger values of *a* after the crossing region, the lowest band is governed by the non-Bragg gap, which exhibits a substantial red-shift with *a*. It is interesting to note that non-Bragg gaps are relatively not so sensitive to longitudinal periodicity, whereas Bragg gaps shift considerably to lower frequencies for a longer pitch length. The central frequencies of the Bragg gaps predicted by the model are traced by the orange, green, and blue solid lines for *p* = 0.6, 0.8, and 1*μ*m, respectively. Again, the traces follow the high reflection bands quite well in the Bragg regime. It is worth noting that this ellipsoid-based method has been widely used to model chiral sculptured thin films (STFs) with columnar structures. We have found that this technique can be generally applied to the present helix array in the Bragg regime in spite of different structural features compared to STFs. Care needs to be taken, though, when the volume fraction of the dielectric constituent is too large, which may deviate from the assumption made under the homogenization approach [39]. Another informative analysis is to examine the dependence of gaps with respect to longitudinal periodicity. According to equation (9), the Bragg wavelength ${\lambda}_{0}^{\mathit{Br}}$ is linearly proportional to the longitudinal period *p*. Figure 3(b) depicts gap regions versus *p*. Structures with *a* = 0.6, 0.8, and 1*μ*m are denoted by yellow, green, and blue, respectively. For a smaller lateral period, e.g., *a* = 0.6*μ*m, the gap is mostly operated in the Bragg regime as illustrated by the yellow band, where the Bragg wavelength displays a linear dependence with the longitudinal period *p*. Non-Bragg gap can only be observed at a short pitch length, appearing in the bottom left corner of the figure. For helices separated further apart to *a* = 0.8*μ*m (the green band) and *a* = 1*μ*m (the blue band), more regions are governed by non-Bragg gaps. Comparing non-Bragg gap to Bragg gap, the former exhibits a noticeable shift in wavelength, whereas the latter only displays a relatively small blue shift. The three solid lines represent ${\lambda}_{0}^{\mathit{Br}}$ in equation (9) calculated by the model for different values of *a*. We can see that the slight blue shift of Bragg wavelength with increased *a* can also be inspected from the calculations, in agreement with our numerical results.

### 3.1.2. Non-Bragg gap regime

In addition to Bragg gaps, origin of forbidden gap has also been discussed by hybridization of two crossing bands. A forbidden gap can open up about the crossing point of two bands owing to the interaction of modes, typically between a propagating wave with a local resonance of the constituent element. The anti-crossing behavior is a fingerprint of strong coupling and has been observed in many physical processes. The prominent feature of the coupling phenomenon is represented by frequency splitting of bands, which can be derived classically from the coupling between resonance states of two oscillators [40].

In the present structure, the interaction involves circularly polarized light with the resonance of the helix array. Different levels of coupling strength between circularly polarized light with the structure can be examined when we vary the helix radius *R* to tune the chirality of the structure. In Fig. 4(a)–4(c), we present the band structures for three helix structures that only differ by helix radius *R*. Figure 4(a) shows the band structure for a rod-like structure. At lower frequencies, the dispersion curve displays a propagation wave characteristic with a proportionality between frequency and the wave vector. Around 460THz, the propagating band crosses with another dispersion curve, which is the resonance band of the rod array. When the helix radius increases to 0.03*μ*m, the structure evolves from a rod-like structure to a more helical geometry. One narrow gap opens up as denoted by the frequency splitting in the yellow region. The abnormal dispersion curves manifest themselves as the anti-crossing of two coupled bands. The frequency splitting is attributed to the coupling between the local resonance of interacting helices with the propagating mode of the optical field. The enhanced structural chirality gives rise to a stronger coupling strength, enlarging the frequency splitting as shown in Fig. 4(b). A larger helix radius (*R* = 0.05*μ*m) further enhances the level of repulsion, yielding a wider band gap as shown in Fig. 4(c). In addition, the anti-crossing of bands is normally accompanied with the transition of mode patterns. The field distributions of modes for the coupled bands in Fig. 4(c) are shown in Fig. 4(d). The mode patterns are monitored at the same plane normal to the helical axis. For the upper band, the mode profiles in point A evolve rapidly after crossing the lowest point to B. Similar behaviors are shown for the lower band from C to D. Such abrupt change of mode patterns is not observed for the bands in the Bragg regime.

The analysis shown in Fig. 4 has revealed that the BC branch is deduced from the eigenmode of the rod structure with the field distributions shown in the upper panel of Fig. 4(e). Upon increasing of *R*, the profiles of the mode patterns are maintained in the transverse plane as shown in the lower panel of Fig. 4(e), the properties of which can be obtained by the eigenvalue equation of the rod array [41]. The coupling between this resonance band with the propagating circularly polarized mode is attributed to the like symmetry of the field distributions [42], which can be examined by comparing their field distributions shown in the upper and lower panels of Fig. 4(f). The eigenvalue equation can be adapted to calculate the cut-off frequencies of the local resonance in the present structure as

*n*is the refractive index of the dielectric constituent,

*k*

_{0}is the wavevector in free space,

*r*is the wire radius,

*S*

_{0}is the first coefficient of the Rayleigh identity, and

*J*

_{0}and

*H*

_{0}are the Bessel and Hankel functions of order zero of the first kind, respectively. To inspect the resonance mode with respect to the gap region, the cut-off frequencies/wavelengths of the resonance mode are traced by the dashed curves as shown in Fig. 3. One can see that the variation of the non-Bragg gap follows well with the dashed curves. Note that the resonances are traced under the cut-off condition, which results in a red-shift of the curves with respect to the band gap regions. It is also worth mentioning that multiple Bragg regimes may exist in the present structure. The non-Bragg mechanism is associated with distinct underlying physics and cannot be obtained by second- or higher order Bragg regimes [43–45].

#### 3.2. Dispersion characteristics in different regimes

When the lowest forbidden band evolves from Bragg gap to non-Bragg gap with increased lateral period, the band structure and the related properties undergo a number of modifications. Crossing of bands reshapes the dispersion curves, which changes the wave propagation properties accordingly. To examine the modifications, we calculate the dispersion surfaces to inspect how equifrequency contours (EFCs) develop in different scenarios. Figs. 5(a) to 5(c) are the EFCs constructed around the frequency range of the lower gap edge for *a* = 0.3, 0.4, and 0.5*μ*m, respectively, featuring the transition from Bragg to non-Bragg regime. At *a* = 0.3*μ*m, a saddle point with zero group velocity at the center of the plot can be identified as the lower edge of the Bragg gap. Along the Γ*Z* direction, a bell-like dispersion is displayed. As *a* increases, the triangular EFCs on both sides of the Γ*Z* axis move toward the Γ point and then merge into rectangular contours as depicted in 5(b) for *a* = 0.4*μ*m. When *a* = 0.5*μ*m, square-like dispersion contours can be observed, in which segments with a zero curvature allow collimation to occur. The shape of the rectangular contours can be tailored by the ratio of lateral period *a* to longitudinal period *p*. An illustration to depict the collimation effect is shown in Fig. 5(d) and the inset is the corresponding numerical simulation, showing the collimation of the beams within 20° at 393THz. Figs. 5(e) to 5(g) are the EFCs around the upper gap edge for *a* = 0.3, 0.4, and 0.5*μ*m, respectively. At *a* = 0.3*μ*m, the center of the plot is the higher band edge of the Bragg gap, and a cone-like dispersion is displayed showing a normal dispersion relation. As *a* increases to 0.4*μ*m, the approaching of the non-Bragg gap induces strong perturbation to the band, giving rise to considerable changes in EFCs. When *a* reaches 0.5*μ*m, the large curvature of the surface, arising from strong coupling of bands, results in additional saddle points on the surface. The corresponding EFCs shown in Fig. 5(g) predict the emergence of negative refraction for axially propagating circularly polarized waves. An example is depicted in Fig. 5(h) with the numerical simulation as the inset, showing a RH Gaussian beam incident at 8° is negatively refracted in accordance with EFC analysis.

#### 3.3. Operation scenarios with respect to geometrical arrangements

Our analysis has revealed that band gaps in a dielectric helix structure may originate from different mechanisms. While band gap due to Bragg resonance is established via the longitudinal periodicity, the non-Bragg gap is more relevant to the helix lateral spacing. Therefore, such 3D configuration provides another degree of freedom to tailor the relative position between the two bands by geometrical parameters without the use of high-permittivity dielectrics. Lastly, we construct a map featuring various scenarios in Fig. 6 to explicitly reveal the effects of helix geometry arrangement on the optical properties. The helix geometry is represented by the helix aspect ratio, *p/R*, and lateral arrangement of helices is adjusted by *a/R*. *r* is fixed to be *R*/3 in the analysis. To simplify the illustration, we define the operation regions by the features of RH gap. For a specific choice of helix geometry, the lowest RH gap for axially propagating light may be categorized as a Bragg gap or a non-Bragg gap, colored in blue and yellow, respectively. The region in between, colored in green, is when the Bragg gap and non-Bragg gap have merged around the same frequency range. In general, Bragg gap is preferably to occur when the helices are closely-packed or with a larger aspect ratio. The operation can be further refined concerning the responsive handedness of light. Within the Bragg gap-dominating regime, the area denoted in light blue represents the cases when the structure only exhibits a RH Bragg gap without a LH gap. The operation is referred to as circular Bragg regime, in which only one circular polarization state shows high reflection [7]. Yet via spatial arrangement, LH motifs may emerge in an otherwise RH helix structure [32], as indicated in the dark blue region. When helices are separated further apart or with a smaller aspect ratio, non-Bragg gap tends to take over. The transition region from Bragg gap to non-Bragg gap is denoted in green, where two bands emerge into a broad high reflection band. For an operation scenario in the lower-right corner, denoted in grey, helices are with a more ring-like geometry and are sparsely arranged. CP wave traveling along the helical axis can no longer couple efficiently with the helix structure. The reflection level drops and the optical response is governed by the scattering fields of helices. To clearly depict each operation regime, we also present the corresponding gap maps along traces **b** to **g** in the diagram, as shown in Figs. 6(b) to 6(g). RH and LH bands with a reflectance above 0.5 are illustrated in orange and blue, respectively, and their overlap regions appear in dark grey. Figs. 6(b) to 6(d) represent the movement of bands when the structure evolves from a squeezed helical geometry to an elongated helical shape. With the increase of the pitch length, more regions are operated in the Bragg regime and the resonance bands move downward to lower frequencies. Figs. 6(e) to 6(g) illustrate the movement of bands when the structure transforms from a densely-packed to a sparsely-packed array. With smaller helix spacing, the structure is predominately operated in the Bragg regime. The increase of the lateral helix spacing shifts the non-Bragg gap to lower frequencies and the interfered region with the Bragg gap can be adjusted by the lateral period. These gap maps explicitly exemplify how the operation scenarios can be manipulated subject to geometrical arrangements of the helix structure. It is also worth noting that the diagram is constructed under the helix radius of *R* = 0.15*μ*m. Although the boundaries of operation regime may vary slightly, we have observed the same transitions between operation regimes for a structure consisting of helices with a different choice of radius.

## 4. Conclusion

In conclusion, we have presented the origin of band gaps in a 3D dielectric helix structure. Band gaps arising from different mechanisms exhibit distinct properties and can be manipulated by geometrical arrangements. We show that for a densely-packed helix array with a large aspect ratio of the helix, the optical properties are governed by Bragg resonance associated with longitudinal periodicity. Other versatile scenarios may arise by increasing the helix period or decreasing the aspect ratio, where non-Bragg gaps attributed to mode coupling interfere with ordinary Bragg gaps. The interplay of gaps with different mechanisms in such 3D helical configuration yields versatile behaviors for circularly polarized light, which may also have implications for the study of optical properties in other 3D chiral periodic structures.

## 5. Appendix

To characterize the optical properties in the Bragg regime, we use the formulation adapted to model chiral sculptured thin film (chiral STF), as the helical inclusion in chiral STF can most resemble our helix structure. In this method, the chiral STF is described by effective constitutive parameters. These effective parameters are estimated by a homogenization process of an ellipsoid-based chiral structure, in which the constitutive relations can be written as [21]

*∊*

_{0}and

*μ*

_{0}are the permittivity and permeability of free space.

*∊̱*is the permittivity dyadic, defined as

_{ref}*∊*

_{a,b,c}are the effective permittivities in the rotating coordinate frame.

**S̱**

*is the tilt dyadic that represents the elevated helix structure above the xy plane and*

_{y}**S̱**

*is the rotation dyadic that accounts for the rotational nonhomogeneity along the helical axis.*

_{z}Based on the homogenization process, the effective permittivities *∊*_{a,b,c} can be obtained by solving the dyadic equation

*f*is the volume fraction of the helix inclusion,

**0̱**is the null dyadic. Dyadics

**a̱**

*and*

_{s}**a̱**

*are given by*

_{f}*∊*and

_{s}*∊*are the electric permittivities of the inclusion and void, respectively.

_{f}**I̱**is the identity dyadic.

**Ḏ**

*and*

_{s}**Ḏ**

*are the depolarization dyadics of the inclusion and void, respectively, and can be calculated given the geometry of the ellipsoid [20]. The central wavelength ${\lambda}_{0}^{\mathit{Br}}$ of the Bragg gap can be estimated as the following where $\tilde{{\u220a}_{d}}={\u220a}_{a}{\u220a}_{b}{\left({\u220a}_{a}{\text{cos}}^{2}(p/(2\pi R))+{\u220a}_{b}{\text{sin}}^{2}(p/(2\pi R)\right)}^{-1}$ and*

_{f}*p*is the pitch length.

## Funding

Ministry of Science and Technology (MOST) in Taiwan (105-2221-E-007-071-MY3, 105-2119-M-007-011.).

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