Group theory of circular-polarization effects in chiral photonic crystals with four-fold rotation axes applied to the eight-fold intergrowth of gyroid nets

We use group or representation theory and scattering matrix calculations to derive analytical results for the band structure topology and the scattering parameters, applicable to any chiral photonic crystal with body-centered-cubic symmetry $I432$ for circularly polarized incident light. We demonstrate in particular that all bands along the cubic $\left[100\right]$ direction can be identified with the irreducible representations $E_{\pm }$, $A$, and $B$ of the $C_4$ point group. $E_{+}$ and $E_{-}$ modes represent the only transmission channels for plane waves with wave vector along the $\Delta$ line, and $E_{-}$ and $E_{+}$ are identified as noninteracting transmission channels for right- and left-circularly polarized light, respectively. Scattering matrix calculations provide explicit relationships for the transmission and reflectance amplitudes through a finite slab which guarantee equal transmission rates for both polarizations and vanishing ellipticity below a critical frequency, yet allowing for finite rotation of the polarization plane. All results are verified numerically for the so-called 8-srs geometry, consisting of eight interwoven equal-handed dielectric gyroid networks embedded in air. The combination of vanishing losses, vanishing ellipticity, near-perfect transmission, and optical activity comparable to that of metallic metamaterials makes this geometry an attractive design for nanofabricated photonic materials.

Figure 1

Construction of the 8-srs by three replication steps. (a) 1-srs: cubic $I{4}_{1}32$ (214); (b) 2-srs: tetragonal $P{4}_{2}22$ (93); (c) 4-srs: cubic $P{4}_{2}32$ (208); (d) 8-srs: cubic $I432$ (211). In each step the number of srs nets is doubled by generating translated copies (blue) of the already existing nets (green). All nets are identical and equal handed. Numbers in parentheses refer to the symmetry group numbers as in Ref. 26.

Figure 2

An alternative construction method that yields the same 8-srs structure shown in Fig. 1. Left: The BCC translational unit cell is obtained by placing degree-three vertices at the midpoints of the hexagonal facets of a Kelvin body. Edges connect the hexagon's center point to every second vertex, such that rotational symmetries of the Kelvin body are maintained. Right: When repeated periodically, the 8-srs consists of 8 equal-handed interwoven srs nets; only one of the eight nets is shown for clarity.

Figure 3

Different cross sections with $\left[100\right]$ inclination through the 8-srs reveal its four-fold symmetry. The parameter $t$ denotes the position of the termination plane within the cubic unit cell. 4-fold rotation and $4_2$ screw axes are marked by the square and barred square symbols, respectively. The gray square represents the cross section of the cubic unit cell whose vertices are located at the $O$ (432) symmetry point marked by a blue square symbol (same choice as in Ref. 26).

Figure 4

Photonic band structure and transmission spectrum of the 8-srs PC. Left: Band structure for $k$ along $\Delta$ (see Fig. 3). The bands are colored according to their symmetry behavior corresponding to the irreducible representations $i\in \{A,B,E_{+},E_{-}\}$ of the $C_4$ point symmetry (cf. Table ). The $E_{\pm }$ bands that can couple to plane waves at normal incidence are further underlaid by dots of size proportional to the coupling constant $\beta$. Dots are smaller than the linewidth for $\beta ⪅0.1$ and hence invisible. See discussion in Sec. 3b for the meaning of the inset. Right: Transmission of light at normal incidence through a quasi-infinite slab of thickness $N_z=53$, $\left[100\right]$ inclination, and the two terminations shown in Fig. 3: $t=0$ or the eyes cut shown in black and $T=0.25a$ or the dog bones cut shown in brown. The spectrum is the same for any polarization state and is illustrated by the thin lines. The thick and more saturated lines represent a convolution with a Gaussian of width $\delta \Omega =0.002$ eliminating the sharp Fano and Fabry-Pérot resonances. The teal points in the transmission spectrum mark the transmission minima at the pseudo–band gap at roughly ${\Omega }_g=0.64$ for a slab of thickness $N_Z=4,\cdots ,53$, respectively.

Figure 5

Band structure of the inverse 8-srs PC for $\varepsilon =1$ within the networks and $\varepsilon =5.76$ outside and volume fraction $\Phi \approx 41\%$ (upper left quadrant in Fig. 6). The PBS is calculated using BCC and simple cubic (SC) symmetry. In the plots labeled “SC” the band structure of the simple cubic lattice is calculated for a grid discretization of $M^3={32}^3$ and unfolded to the BCC $\Delta$ path (Ref. 73). The bands are colored by their respective symmetry behavior corresponding to $i=\{A,B,E_{+},E_{-}\}$. Some of the interesting regions around $H$ and $\Gamma$ are magnified to demonstrate the excellent agreement between theory and numerics in the SC calculations. Plots labeled “BCC” use a BCC Fourier grid that does not maintain all symmetries of the BCC symmetry; see Appendix . The BCC calculations illustrate that the violation of correct symmetry lifts the degeneracy of the bands at 3-fold degenerate points, even for a fine grid with $M=64$.

Figure 6

Relative gap width ${\delta }_G=2({\omega }_2-{\omega }_1)/({\omega }_2+{\omega }_1)$ of the symmetry-induced band gap of midgap frequency ${\Omega }_G$; ${\omega }_1$ is the maximum frequency of the two fundamental bands of character $E_{+}$ and $E_{-}$, respectively, and ${\omega }_2$ the minimum of the two air bands of the same character. Left: Color-coded map of ${\delta }_G$ as a function of volume fraction $\Phi$ and dielectric contrast $\psi ={\varepsilon }_{8-\mathrm{srs}}:{\varepsilon }_{\mathrm{background}}$ on a logarithmic scale. The join frequency ${\Phi }_J\approx 38\%$ indicates a topological change: For $\Phi <{\Phi }_J$, the individual srs nets are disconnected; for $\Phi >{\Phi }_J$ they overlap to form a single connected component. The orange point “x” marks the choice of parameters for Fig. 4. Right: Gap map along the line indicated by the white arrow on the left including the x point. The frequency is scaled by the effective index $n_{\mathrm{eff}}:=\sqrt{{\sum }_m{\Phi }_m{\varepsilon }_m}$ with $m\in \left\{8-\mathrm{srs},\mathrm{background}\right\}$ on the ordinate. The position of the band gap hence does not depend on the choice of the absolute value of dielectric constants. It turns out that the x point is close to a band structure topology change at $\Phi \approx 31\%$ and $\psi \approx 6.5$ where the $T_1$ (black curve) and $T_2$ (green curve) points at $H$ are accidentally degenerate.

Figure 7

Brillouin zone of the BCC lattice: A rhombic dodecahedron whose edges are illustrated by yellow bars. The irreducible BZ ($1/48$ of the BZ due to 24 rotations in $O$ (Table ) and another 24 time-inverted rotations) is the pyramid framed in orange. The $\Delta$ line marked in blue is one of its edges and connects the $\Gamma$ point in the origin with the $H$ point that is a 4-connected vertex of the BZ at $2\pi /a·{(1,0,0)}^T$.

Figure 8

Illustration of the four mode structures of a monochromatic vector field that spatially transform according to the respective irreducible representation of $C_4$ (cf. Table ). The field is shown at an arbitrary point in time at four symmetry-equivalent points. For the $E_{\pm }$ modes for which the characters are not all real, the transformation depends on the polarization state and we split each mode into a red RCP part with temporally right rotation as seen from the receiver and a blue LCP part rotating opposite. The $A$ and $B$ profiles do not couple to a plane wave (center) as opposite contributions with a phase shift of $\pi$ cancel out. The RCP part of an $E_{+}$ mode and the LCP part of an $E_{-}$ mode cannot couple to a plane wave for the same reason.

Figure 9

Simulated circular dichroism (CD) and optical activity (OA), defined in Eq. (1), and transmissivity $T_{\pm }$ in both transmission channels $E_{\pm }$ for the reflection and transmission of a plane wave at normal incidence: The 8-srs PC slab has termination $t=0.25a$ and thickness $N_z=4$. Since optical experiments cannot measure phase differences $>\pm {90}^{\circ }$, OA is wrapped onto the interval $[-{90}^{\circ },{90}^{\circ }]$ by $\mathrm{OA}\mapsto \arctan \left[\tan \right(\mathrm{OA}\left)\right]$.

Figure 10

Diagrammatic representation of the processes that lead to different circular-polarization properties of reflected and transmitted light. Assuming that there is a single propagating mode in each channel and direction (a) at a given frequency ${\omega }_0$, transmission (b) and reflection (c) can be understood by considering positive, upwards pointing $k$ ($\uparrow$) and their counterparts for negative $k$ ($\downarrow$). If the modes further have a leading Bragg order in propagation direction (i.e., interference may be neglected) and the thickness of the slab $L$ is an integer multiple of the lattice parameter $a$, the difference in $k$ between two modes $\delta k$ corresponds to an optical phase shift $\delta \phi =\delta kL$ [cf. Eq. (2)]. The regions labeled upper (U) and lower (L) interface represent the interfacial two-dimensional planes between free space and the photonic crystal that are inflated to finite thickness for visualization. The sign and coloration represents the $E_{\pm }$ channels in the same manner as in Figs. 4, 8, 5, and 9.

Figure 11

Incompatibility of planar hexagonal and spatial BCC symmetry with the discrete Fourier grid used in PBS plane-wave-based frequency domain eigensolvers. For any planar object with hexagonal symmetry, a discretized representation using a fast Fourier grid that takes exactly two linearly independent basis vectors cannot maintain the full hexagonal symmetry, as (a) illustrates for a circle; a discretization by hexagons (reduced Wigner-Seitz cells), as shown in (b), would alleviate this problem but cannot be implemented for the Fourier analysis. (c) The BCC case represents a 3D analogon with the same problem. The Brillouin zone (solid cell) is a rhombic dodecahedron with full $O$ symmetry, yet the parallelepiped formed by the basis vectors of the Fourier grid representation (hollow cell) only maintains $D_3$ symmetry with a single 3-fold axis (red) and three 2-fold axes (cyan).