A spiral is a type of three-dimensional chiral structure. This kind of structure has attracted attention because of its optical activity [1] and because of its geometric resemblance to the structure of diamonds [2]. Recent studies [3,4] have shown that spiral-structured photonic crystals possess complete photonic bandgaps and they are also candidates for negative refraction materials [5]. A few innovative techniques have been used to fabricate various forms of spiral structures on the micron and sub-micron scales; examples of such techniques are glancing angle deposition (GLAD) [4], two-photon processes [6,7], and holography-lithography [8]. We have found through computer simulations that there are at least two ways to produce periodic arrays of separated hollow spiral structures (spring-like structures with holes in their axes, see Fig 2) using the holographic lithography method. Spring-like spiral structures containing metallic nano-Ag can also be made by two-photon writing processes [7]. The spirals fabricated by holographic lithography are separated and hence their properties are expected to be different from those proposed by Toader and John [3]. In this paper, we describe the band structure and transmission properties of a periodic array of spirals and show that they possess significant polarization gaps in which one circular polarization is forbidden.

All the band structures described here were calculated by the plane-wave expansion method. We used 2000×2000 matrices and, when we doubled the matrix size, there were no observable differences to the naked eye in the band structures in the frequency range of interest. We calculated the transmission spectra by using the scattering matrix method (SMM) [9,10]. The plane wave method was first used to study the spiral structure proposed by Toader et al. [3] and our band structure results showed a perfect match with that earlier spiral structure [3]. The spiral structure (with ε=11.9) fabricated by the GLAD method exhibits robust photonic gaps. The transmission calculation for incident light along the Γ to Z direction for circularly polarized light in the Toader et al. structure is shown in Fig. 1. Although the basic building block of their structure is spiral, the transmittances for right hand (RH) and left hand (LH) circularly polarized light are almost the same. This weak chiral character is probably because the spirals are strongly overlapping. To verify this, we studied systems that have the same basic building blocks as that proposed by Toader et al. [3], i.e, spirals with square projections. We found that when the ratio between the length of the edge of the square projection and lattice constant is reduced, causing the spirals to become separated, the polarization gap opens up while the absolute photonic band gap disappears.

 

Fig. 1. Transmission spectra for a 16-layer slab of the dielectric structure proposed by Toader [3] for (a) LH and (b) RH circularly polarized incident plane waves.

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We found that well-separated spirals exhibit much stronger chiral character. Such structures can in fact be made using holographic lithography techniques. For example, a 7-beam configuration with 6 equally-spaced circumpolar linear polarized side beams and a circularly polarized central beam will form spirals that are separated and that form a hexagonal periodic array in the x-y plane. This structural geometry has been realized experimentally [11]. We calculated the band structure for a structure consisting of RH spirals, as shown in Fig. 2. This is an idealized structure derived from the one described in [11]. We focused on the ΓA direction, which is parallel to the spiral axis (equivalent to the ΓZ direction in the previous case). The major difference between our structure in Fig. 2 and the one in Ref. [3] is the connectedness between the spirals. The spirals in the structure shown in Fig. 2 are separated and have a smaller filling ratio of 15.2% than the spirals considered by Toader et al. [3].

 

Fig. 2. A schematic diagram of the spiral structure and its Brillouin points. (a) is the side view, (b) is the top view of the array and (c) gives the Brillouin points. The ΓA direction is along the z-axis. The spiral has a dielectric constant of ε= 9. The filling ratio of the structure is 15.2 %.

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The band structure of the spiral structure and the corresponding transmission spectrum through a stack of 16 layers are shown in Fig. 3. In the SMM calculation, each unit cell is divided into 32 sub-layers in the z-direction. We found a sizable polarization gap in the band structure (Fig. 3(c)) and the polarization gap width to the middle gap ratio was 26%. The isolated spirals can be kept in place by dielectric plates above and below the spirals [11]. Alternatively, one horizontal plate per unit cell (vertical distance between plates=a), can be added in the structure. The effect of adding these thin plates (thickness = 3a/128,), where a is the pitch of the spiral in the z-direction, on the band structure is shown in Fig. 3(d). The effect of these thin plates on the polarization gap is very small, except for the overall reduction in frequency (corresponding to a slow wave speed) since there is now more dielectric material in the unit cell, leading to a higher effective dielectric constant. From the transmission calculated for the circular polarized light in Figs. 3(a) and (b), we can deduce that the eigenmode inside the polarization gap is largely LH polarized. Here, we adopt the following geometrical definition of circularly polarized light. At any fixed instant of time, if the tip of the electric field of a plane wave traces a RH helix in space, the plane wave is said to be RH polarized and vice versa. This definition makes it easier to understand the physics of the polarization gap since the polarization gap is produced by the spatial variation of the spiral structures. There is also a narrower polarization gap that forbids propagation of LH polarized light in the ΓM direction, as marked by the shaded area in Fig. 4c. We note that various chiral structures leading to some form of polarization gap have also been realized experimentally using vacuum deposition [12], cholesteric liquid crystal structures [13, 14], and chiral optical fibers.

The properties of the eigenmodes of an infinite crystal can also be characterized by examining the field patterns obtained using the plane wave method. The dominant Fourier component of the Bloch wave was first identified. In a periodic system, the Bloch eigenmodes specified by a k vector have infinitely many k+G (G is a reciprocal lattice vector) components and plotting the band structure in the usual reduced zone scheme does not give information about the magnitude of these components. Identifying the dominant Fourier component gives some information on the strength of the Bragg scattering and whether or not the wave travels with a positive group velocity. Figure 4 shows the magnitude of the Fourier components (k+G) for the Bloch modes in the ΓA direction. The sizes of the symbols are proportional to the magnitude of the plane-wave components in the eigenmode. We see from Fig. 4 that a single Fourier component dominates Bloch wave and most of the Fourier components are too weak to be seen in Fig. 4(a), while they are slightly more conspicuous in Fig. 4(b). The figure indicates that the band folding at the edge of the Brillouin zone is very weak and that the wave travels with positive group velocity up to about ωa/c=0.7. Because of the existence of a single dominant Fourier component for most of the k-points, it is more instructive to discuss the dispersion in an extended-zone scheme.

 

Fig. 3. Transmission spectra and the band structure of the RH spiral structure shown in Fig 2. The incident plane wave is LH and RH circularly polarized in (a) and (b), respectively. The band structure is shown in (c). The transmission spectrum is calculated for 16 layers in the z-direction and the shaded areas mark polarization gaps. In (d), thin plates of a dielectric material are added in each period in the z-direction. Bands “a” and “b”, of different polarizations, are marked by red and blue colors, respectively.

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Fig. 4. Band structure along ΓA plotted in the extended zone scheme with the magnitude of the plane-wave components indicated by the sizes of the symbols. Panel (a) is for the structure with continuous spirals and (b) is for the structure with thin plates inserted in each period. Bragg scattering is barely observable in (a). The first Brillouin zone is the region bounded between ±0.5 (in units of kπ/a). Bands “a” and “b” are marked by circles and crosses, respectively.

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The eigenmodes are analyzed in two ways. We first consider the ratio $\frac{〈H_x〉}{〈i{H}_y〉}$, where the spatial average, 〈…〉, is taken inside a unit cell. This method gives a good effective representation for low-frequency eigenmodes. The real parts of the ratios are shown in Fig. 5. The imaginary part is very small in this frequency range. The results show that, for the RH spiral structure shown in Fig. 2, band “a” is essentially RH polarized, while band “b” is LH polarized, and the eigenmode inside the polarization gap is strongly LH polarized, consistent with the transmission calculations.

 

Fig. 5. The ratio between different components of the H-field in the (right-handed) spiral structure of Fig. 2. The ratio is expected to be 1 (-1) for right- (left-) handed circularly polarized waves.

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A second way of characterizing the wave is to find the coupling magnitude of the eigenmodes with a plane-wave that has well-defined circular polarization. The coupling coefficients are defined as $C_{\pm }\left(z_0\right)={\mid \int \int \frac{1}{\sqrt{2}}\{{\left(\hat{x}\pm i\hat{y}\right)}^{*}•\overrightarrow{H}\left(x,y,z_0\right)\}\mathrm{dxdy}\mid }^2$, where H→(x,y,z₀) is the field of an eigenmode in a plane at z=z₀. C₊ and C^- are, respectively, the coupling coefficient of the left- and right-handed polarized plane waves to the normal mode, and the value depends on z₀, which is varied to obtain the upper and lower bounds of $\frac{C_{-}}{C_{+}}$ the- ratio. If the ratio is very large or very small, the eigenmode is circularly polarized. In the calculation, we considered 128 values of z₀. The results are shown in Fig. 6. The ratio is calculated along band “a” and band “b” and the results are consistent with those shown in Fig. 5. The above analyses are also consistent with results obtained from the SMM calculations. As a control calculation, this analysis was also applied to a structure with cylinders of circular cross-sections put in a triangular lattice. The results were as expected: $\left(\mid \frac{C_{-}}{C_{+}}\mid ~1\right)$, i.e., the eigenmodes are linearly polarized in non-chiral structures.

 

Fig. 6. Upper bounds (open circles) and lower bounds (crosses) of the coupling coefficient ratios of a circular-polarized plane-wave with eigenmodes of spiral structures. Panels (a) and (b) show the coupling coefficients for a RH and LH dielectric spiral, respectively.

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The eigenmode characterization shows that the polarization gap for the structure in Fig. 2 will impede the propagation of right-handed circularly polarized light. This can be explained by the fact that right-handed circularly polarized light with a wavelength that matches the pitch of the spiral will match the symmetry of the right-handed spiral structure. As a result, there are two eigenmodes at that wavelength, one with a higher frequency from concentrating its electric field in the air, while the other localizes its electric field mainly in the dielectric material and has a lower frequency. A polarization gap is then formed. It is not geometrically possible for LH polarized light to “follow” the RH spiral and therefore there is no gap. From the results shown in Figs. 3(a), 3(b), 5 and 6, we expect that if an unpolarized beam is incident to this spiral structure along the ΓA direction, it will be separated into one RH polarized beam and one LH polarized beam. Inside the polarization gap, the LH polarized beam will be transmitted through the spirals and the RH beam will be reflected. Consequently, this structure behaves as a polarization filter.

We also examined the polarization gap as a function of k_//, the k-vector component parallel to the x-y plane. The results are shown in Fig. 7 with k_// along ΓM and ΓK. The figure indicates that the polarization gap is fairly robust to incidence angles.

 

Fig. 7. The polarization gap as a function of k_//. The region between the two curves is the frequency range in which only a single polarization is allowed inside the spiral structure. The dashed lines represent k_// =ω/c.

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In summary, we have performed band structure and transmission calculations on the dielectric spiral photonic crystals depicted in Fig. 2. We found that photonic crystals with disjoint spirals exhibit significant polarization gaps in which only one circular polarization is allowed. We analyzed the eigenmodes and results are consistent with transmission calculations. We proposed laser interference configurations that can achieve such geometries. These geometries have been demonstrated experimentally [11].

This work was supported by Hong Kong RGC through 600403.