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Chiral structures exhibit strong interactions with circularly polarized light, and have been demonstrated to show many polarization-dependent properties. Various chiral structures exhibit some level of circular dichroism, where right-handed and left-handed circularly polarized waves experience different transmission. In this study, we use a dielectric helix array as a model system to examine the interactions of circularly polarized light with helical structures. Our results show that circular polarization band gaps can be formed in a dielectric helix array not only by light having the same handedness with the structure but also by light with the opposite handedness, resulting from additional chiral motifs induced by the arrangement of helices. Dual polarization band gaps can thus be tailored by varying the geometrical parameters, and circular-polarization dependent properties can be manipulated for optoelectronic devices and applications.
© 2015 Optical Society of America
1. Introduction
Circular dichroism (CD) is commonly observed in many chiral media, where right-handed (RH) and left-handed (LH) circularly polarized light exhibit different transmission when passing through the media. Artificial periodic structures have been shown to demonstrate strong circular dichroism, giving rise to polarization stop bands that impede one of the two circular polarizations to propagate as illustrated in Fig. 1. The circular polarization-dependent properties have been reported in spiral dielectric photonic crystals [1–3], metallic helix photonic metamaterials [4, 5], stacked achiral planar structures [6], chiral networks [7], etc. The chiral effect in these artificial structures is stronger by orders of magnitude than that in natural chiral media, enabling many interesting applications such as polarization rotators [8] and beam splitters [9].
Fig. 1 An illustration depicts circular dichroism induced by the chiral structure, where right-handed (RH) and left-handed (LH) circularly polarized light, denoted in red and blue, exhibit different transmission when passing through the media.
In dielectric chiral structures, the resonance behavior can be interpreted by circular Bragg phenomenon as seen in cholesteric liquid crystals [10]. At resonance, circularly polarized light with the same sense of rotation with the molecular ordering is reflected within a certain frequency range, forming a polarization stop band. At the lower edge of the stop band, the polarization of light is parallel with the director, which has a higher value of refractive index. For the upper edge, on the other hand, the polarization of light is perpendicular to the director, which experiences a lower value of refractive index. Within the polarization gap, the field decays exponentially and does not contribute to the density of state. Such polarization gap has been reported in many chiral structures [1,6,7,9,11,12], where circularly polarized (CP) light interacts strongly with the chiral structure having the same sense of rotation. Interestingly, many chiral structures also show resonances with CP light that has opposite sense of rotation with the chiral structures [2, 3, 13]. In order to manipulate the optical properties, it is of great interest to study how CP light interacts with the chiral structures and understand the resonance characteristics.
In this study, we use a dielectric helix array as a model system to investigate the propagation of CP light in chiral structures. The fields of CP light follow a spiral route in space, so helical structures are geometrically ideal for studying the resonance of CP light. Various advanced nanofabrication platforms also support real implementations of helix arrays, including glancing angle deposition [14, 15], holographic lithography using circularly polarized beams [16, 17], direct laser writing (DLW) [2], stimulated-emission-depletion DLW [18], and focused ion/electron beam induced deposition [19, 20]. Furthermore, our study also has implications for analyzing the optical properties in other chiral structures that have geometrical elements of both handednesses [6, 7, 21]. In the following, the band structure of the dielectric helices is calculated to analyze the eigenmodes and polarization characteristics. The electric field and energy distributions of the resonance modes are examined and the general properties are discussed. We also analyze the geometry of the helix array, in which the coexistence of helical structures with both senses of rotation can be observed. We find that additional chiral motifs with opposite sense of rotation can be formed by neighboring helices, giving rise to circular polarization band gap with opposite chirality. To examine the dependence of the properties of polarization stop bands on helix structures, the geometrical parameters of the helix array are varied to show the evolution of band gaps. Finally, polarization-dependent properties of the helix array are discussed based on different structural arrangements.
2. Numerical calculation setup
The helix array under study is illustrated in Fig. 2(a). The structure consists of RH helices laterally arranged in a square array as shown in Fig. 2(b). The geometrical parameters are defined by helix radius R, wire radius r, pitch length p, pitch number N, and lattice constant a. The structure is analyzed by tetragonal symmetry and the points within the Brillouin zone are illustrated in Fig. 2(c). We focus our analysis along ΓZ direction, i.e., along the axis of the helix (z direction). We implement the full-field numerical calculations using the 3D software package, Lumerical FDTD Solutions based on finite-difference time-domain (FDTD) method [22]. Periodic boundary conditions are imposed in x and y directions. Unless otherwise specified, the optical properties are calculated with helix radius R = 0.3a, wire radius r = 0.1a, pitch length p = a, and pitch number N = 10. The refractive index of the dielectric is set to be 2.5.
Fig. 2 (a) is an illustration of the dielectric RH helix array with geometrical parameters defined by helix radius R, wire radius r, pitch length p, and lattice constant a. (b) is the top view of the helix array, and (c) shows the points within the Brillouin zone.
To evaluate the chirality of modes, similar definitions in [1,7] are adopted in our simulation. The CP component of the mode is evaluated by normalized CP coupling coefficients, ηRH and ηLH, given by the overlap integrals between the magnetic field of Bloch modes at z = z0 plane and the CP waves as
where ηRH and ηLH are the normalized CP coupling coefficients of RH and LH circularly polarized waves, described by and , respectively. The value of ηRH and ηLH have been normalized to total energy and vary slightly depending on the choice of z0. In our simulation, we chose the z-normal plane z0 = 0.5a along z direction. Circular dichroism index (CD index), a measure of the degree of RH or LH circularly polarized mode, is defined by where sgn is the sign function and is the wave vector of the incident CP wave. By this definition, a pure RH circularly polarized mode has a CD index of +1, whereas a pure LH circularly polarized mode has a CD index of −1. The efficiency of a plane wave to be coupled into the Bloch mode is evaluated by the coupling index κ = ηRH +ηLH. Since RH and LH circularly polarized modes are orthogonal, the coupling index would be 0 ≤ κ ≤ 1. A small value of the coupling index implies a low coupling efficiency of the mode. For simplicity, the notations of RH circularly polarized light(mode) and LH circularly polarized light(mode) are denoted hereafter as RH light(mode) and LH light(mode), respectively.3. Results and discussions
Figure 3(a) is the band structure of the helix array with a helix radius R of 0.38a. The corresponding CD index of the mode is represented by different colors. RH modes with a CD index around +1 are denoted in red, LH modes with a CD index around −1 are denoted in blue, and modes with a CD index close to zero are in green. The coupling index κ of the mode is indicated by the size of the symbol. The band structure shows that no significant chiral resonance is observed at low frequencies below the first Brillouin zone, and the CP light only exhibits strong resonance with the structure at the edge of the second Brillouin zone. This implies that strong resonance occurs when the wavelength of light matches the pitch length of the dielectric helix [1, 2]. The resonance gives rise to a forbidden band of RH light. The corresponding reflectance spectrum for RH light is shown in Fig. 3(b), where a RH polarization gap (RH gap) with high reflection can be observed. Within the gap, LH light can still propagate, resulting in a low level of reflectance of LH light as shown in Fig. 3(c). As frequency increases, LH light also forms a polarization gap (LH gap), consistent with the high reflection region of LH light in Fig. 3(c). In this example, one can see that the polarization gap can be found for both RH and LH light, even the structure consists of only RH helices. A complete photonic band gap is then formed in the frequency range where two polarization gaps overlap as denoted by the gray region.
Fig. 3 (a) Band structure of the helix array along ΓZ direction. The CD index of the mode is represented by different colors and coupling index is denoted by the size of the symbol. (b) and (c) are the reflectance spectra for RH and LH light, respectively.
We first examine the interaction of RH light with RH dielectric helices. Previous studies have suggested that a circular polarization gap can be formed for CP light with a wavelength that matches the pitch of the helix [1]. To examine the resonance, we monitor the eigenmodes at both sides of the RH gap. The electric field distributions at the low and high frequency edges are monitored along the axis of the helix as shown in Figs. 4(a) and 4(b), respectively. The direction of the electric field is denoted by the arrows. For the mode at the low frequency edge of the RH gap, the electric field follows the dielectric spiral, in which the energy is concentrated in the region with a higher value of refractive index. For the mode at the high frequency edge, the direction of the electric field rotates in space and is perpendicular to the dielectric spiral. The energy is mostly located in air and thus results in a lower value of effective refractive index.
Fig. 4 The electric field distributions of the RH gap at the (a) low frequency edge, and (b) high frequency edge along the axis of the helix. The direction of the electric field is denoted by the arrows. (c) An illustration shows that LH motifs denoted in blue can be constructed by segments of adjacent RH helices. The inset shows the locations of the LH motifs in a RH helix array. Effective LH structures are denoted in blue. (d) The electric field distributions of the LH gap at the low frequency edge, and (e) high frequency edge along the axis of the helix.
As there is no obvious LH structure in RH helices, it is less intuitive to understand the formation of LH polarization gap. Nevertheless, the chiral structure should consist of LH motifs that can interact strongly with LH light. We first analyze the geometrical features of a RH helix array. The geometrical analysis reveals that LH motifs can be constructed by segments of adjacent RH helices, where LH structures are embedded between RH helices. Figure 4(c) illustrates the RH helices in orange and the embedded LH motifs in blue. The inset figure shows the relative positions of the LH motifs between RH helices. While RH light follows a continuous RH helical structure at resonance, the resonance of LH light is built up in fractional LH structures. It implies that the resonance properties of RH modes are subject to the change of helical shape, whereas the resonances of LH modes are built more relying on the spatial organizations of symmetric elements [23]. The field patterns at the low frequency and high frequency edges of the LH gap are monitored along the axis of the helix as shown in Figs. 4(d) and 4(e), respectively. One can see that the directions of the electric fields for both LH modes follow a LH route and are mutually orthogonal.
Since the circular polarization gap is closely related to the geometrical arrangement of the structure, the properties of the (polarization) band gaps can be manipulated by varying the structural parameters. To exemplify the effect, we vary the helix radius R and wire radius r, and examine the evolution of RH and LH polarization band gaps. Figure 5(a) shows the shifts of the low frequency edge (Ωl) and high frequency edge (Ωh) of the RH and LH polarization band gaps while varying the helix radius R from 0.14a to 0.5a. Ωl and Ωh are defined by the corresponding frequencies when the reflectance drops to 0.8 in order to keep track of the high reflection band in the spectrum. The RH and LH bands are depicted in red and blue, respectively. At R = 0.14a, a RH gap is observed at Ω =0.7578 to 0.8283. Since RH helices are separated apart due to a small value of R, LH structures between adjacent RH helices can not effectively support the resonance of LH light and hence no obvious LH gap is produced. When R increases, the LH gap emerges. At R = 0.2a, both RH and LH gaps are observed, and the corresponding spectrum is shown in Fig. 5(b). When R increases to 0.3a, Ωh of the RH gap is roughly at the same frequency with Ωl of the LH gap. It implies that the RH gap is neighboring to the LH gap, as shown in the corresponding spectrum in Fig. 5(c). While R keeps increasing, part of the RH gap overlaps with the LH gap. Therefore, one can find that a complete gap is formed and surrounded by a RH gap and a LH gap toward the low and high frequencies, respectively. Figures 3(b) and 3(c) are the spectra of this scenario at R = 0.38a. At R = 0.46a, both RH and LH gaps have a similar Ωh. The corresponding spectrum is shown in Fig. 5(d). One can see that the frequency range of the LH gap has been covered within that of the RH gap, and thus part of the spectral range of RH gap shows a complete band gap denoted in gray.
Fig. 5 Shifts of the low frequency edge (Ωl) and high frequency edge (Ωh) of the RH and LH polarization band gaps while varying the helix radius.
Figure 6(a) shows the shifts of Ωl and Ωh of the RH and LH polarization band gaps while varying the wire radius r from 0.02a to 0.3a. For a thin wire radius below 0.02a, neither RH nor LH light exhibits strong resonance and therefore there is no stop band for either polarization. As r increases, both RH and LH gaps emerge. At r = 0.08a, the LH gap is neighboring to the RH gap, as shown in the spectrum in Fig. 6(b). As the wire radius keeps increasing, the LH gap gradually shifts toward the RH gap, in which a complete band gap can be formed between the RH gap and the LH gap. The full LH gap moves into the RH gap at r = 0.2a as shown in the spectrum in Fig. 6(c), and it keeps shifting toward low frequencies with increase of the wire radius. At r = 0.28a, the LH gap is located in the middle of the RH gap. In this case, the complete band gap, denoted in gray, is situated in the same frequency range as the LH gap and is surrounded by RH gaps in both sides as shown in the spectrum in Fig. 6(d).
Fig. 6 Shifts of the low frequency edge (Ωl) and high frequency edge (Ωh) of the RH and LH polarization band gaps while varying the wire radius.
From the analysis results, one can see that as long as the wire radius is thick enough to support the resonance of RH light, a RH gap can be easily formed in RH helices. The increase of the helix radius R slightly shifts Ωl of the RH gap toward low frequencies, whereas Ωh is roughly at the same frequency. This is consistent with the properties of the RH modes described in Figs. 4(a) and 4(b). For the eigenmode at Ωl, fields are concentrated in the dielectric regions. Relatively more energy is confined in regions for a larger R, resulting in the red-shift of the mode. On the other hand, for the eigenmode at Ωh, energy is mostly located in air. The variation of the helix radius thus causes less effect on the resonance frequency. For both Ωl and Ωh of the LH gap, the frequency shifts toward lower frequencies as the helix radius increases. It implies that the increase of the helix radius results in a larger volume fraction of the dielectrics for constructing the fractional LH structures, causing higher effective refractive indices of the LH modes. The variations of the wire radius r induce even larger frequency shifts of the bands. This is attributed to the fact that the volume fraction of the dielectric is proportional to the square of the wire radius. The increase of the wire radius rapidly increases the volume fraction of the dielectrics, thus giving rise to the pronounced red-shifts of the resonances. While comparing the resonance characteristics of RH and LH modes, the resonance frequencies of LH modes exhibit a more consistent scaling behavior with the geometrical parameters. Furthermore, RH modes exhibit much higher light conversion to the opposite handedness due to the change of helical shape, whereas LH modes constantly maintain a high extinction ratio.
Lastly, it is worth noting that the organization of RH helices is important in the properties of LH gaps, as embedded LH motifs are responsible for the formation of LH gaps. Depending on the relative phase and position between neighboring helices, additional chiral structures can be induced. For the case where adjacent RH helices are arranged by a phase shift of 180◦ [3] as illustrated in Fig. 7(a), effective LH structures, denoted in blue, are built up by segments of four surrounding helices. The inset figure shows the top view with the effective LH structures in blue. In such an organization, segments of RH helices can construct LH motifs that well resemble LH helices. One can tailor the parameters to make RH helices and the induced LH structures have a similar geometrically arrangement. Consequently, the RH gap and LH gap can be formed around the same frequency range as an example shown in Fig. 7(b) with R = 0.36a and r = 0.16a. It yields a large overlap of the polarization gaps, which also implies that a larger band gap can be produced based on this arrangement. For a hexagonal periodic array of RH helices in [1], our geometrical analysis reveals that LH motifs can also be constructed by adjacent RH helices. However, the induced LH structures are distorted and the arrangement of segments is not suitable to support the resonance of LH light. Therefore, only one circular polarization gap is observed.
Fig. 7 (a) RH helices are arranged with a phase of 180◦. The induced LH structures are denoted in blue. The inset shows the top view. (b) The reflectance spectrum that shows a large overlap of RH gap and LH gap, resulting in a wide band gap as denoted by the gray region.
4. Conclusion
In conclusion, we present dual circular polarization band gaps based on dielectric helix meta-materials. Additional chirality may be induced by adjacent structures, giving rise to the formation of circular polarization gap with opposite handedness. The polarization gaps can be tailored by varying the geometrical parameters, resulting in various scenarios in the optical properties. Organizations of helices also play an important role in forming the polarization band gap. With proper arrangements of relative position and phase between neighboring helices, circular-polarization dependent properties can be manipulated for optoelectronic devices and applications.
Acknowledgments
The authors would like to acknowledge technical support from Dr. Y. C. Na. This work is supported by Ministry of Science and Technology (MOST) in Taiwan under the following research contracts: 102-2221-E-007-116-MY3 and 103-2633-M-007-001.
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