According to Landau-de Gennes formalism, the free energy of a liquid crystal medium can be written as a function of order parameter tensor, Q _ij in terms of temperature dependent bulk energy (f _T ), Frank’s elastic energy (f _E ) and surface-induced energy written in Rapini-Papoular form (f _S ) as shown below [38]. f = 1 2 a T − T N I * Q i j Q j i + 1 3 B Q i j Q j k Q k i + 1 4 C Q i j Q j i 2 + 1 2 W Q i j − Q ° i j 2 + 1 2 K ∂ Q i j ∂ x k ∂ Q i j ∂ x k (1) where a, B and C are temperature dependent constants, T N I * is the nematic-isotropic transition temperature, W is the surface-induced anchoring strength, K represents the Frank’s elastic constant under single elastic constant approximation (K ₁₁ = K ₂₂ = K ₃₃) and Q _ij ° is the surface-preferred order. The total free energy f is minimized using Euler-Langrangian formalism and explicit finite difference scheme is employed to arrive at the stable configurations.

Periodic arrays of cylinders (radius a = 1 μm, thickness t = 300 nm) and cylindrical rings (inner radius r = 600 nm, outer radius R = 1.0 μm and thickness t = 300 nm) made of dielectric material TiO₂ with periodicity p = 2.5 μμm are placed on the bottom glass substrate coated with ITO as shown in the Figure 1. DFLC mixture is introduced in the cell of thickness d = 2.5 μm. The unit cells of cylindrical and cylindrical ring periodic arrays with lattice constant 3 μm are shown in Figures 2A,B,D,E. At low frequency f _L (“ON” state), the equilibrium director configurations exhibit vertically aligned configuration as shown in Figures 2A,B) for cylinders and cylindrical rings, respectively. Microstructures induce distortion in the surrounding director field leading to suppression of the orientational order, S and hence variation in the optical properties as shown in Figures 2A–E). In the case of complex geometry like cylindrical rings, the director field is also influenced within the inner ring as shown in the Figures 2B,E). Inset in Figure 2E) shows that the director orients parallel to the walls of the inner rings to reduce the elastic energy. The layer-wise director angles-polar angle θ and azimuth angle ϕ averaged in each layer along the Z-direction, n ̄ ( x , z ) are plotted as a function of layer number in the Figures 2C,F). In the “ON” state, LC molecules undergo maximum distortion due to contradicting aligning conditions- 1) by the external voltage induced vertical alignment and 2) the surface anchoring at the glass and microstructures. The plot of n ̄ ( x , z ) as a function of the layer number, Figure 2C), shows decrease in value of θ from the surface induced planar alignment at bottom substrate till the height of microstructure (indicated by the shaded region) after which θ = 0. Azimuthal angle ϕ increases from 0° (X-direction) at the bottom substrate to 90° (Y-direction) at the top substrate giving rise to twisted structure. From Figure 2C) it is clear that distortion in θ and ϕ is more pronounced and complex near the cylindrical rings. In the case of planar alignment (“OFF” state) θ and ϕ undergo maximum distortion near the microstructures (shaded region) from X-axis (θ = 90°, ϕ = 0°) at bottom and top substrates due to the geometry-preferred alignment as shown in the Figure 2F). The distortion is higher for cylindrical rings compared to the cylinders as in the case of “ON” state.

Once the equilibrium director configurations are obtained from free-energy minimization method, we investigate the light-matter interactions of the microstructure embedded LC cell using commercial software CST Microwave studio. The electric and magnetic fields are discretized on a cubic lattice and full wave electromagnetic calculations using FEM method are performed to investigate the electromagnetic response of the microstructures present in the LC cell. Layer-wise averaged director orientations of the LC, n ̄ ( x , y ) are considered for the electromagnetic calculations along the thickness of the cell.

Dual frequency liquid crystal (DFLC) is filled within the glass substrates coated with ITO. A periodic array of TiO₂ structures is placed on the glass substrate as shown in the Figure 2A. The glass substrates induce planar alignment with weak anchoring strength, W = 10⁻⁵ N/m ² along X-axis and the TiO₂ structures induce random anchoring of strength W = 10⁻⁴ N/m ². Standard DFLC mixture CPEP (3F)-5NCS [31] which exhibits a birefringence, Δn = 0.224 at 25°C and γ/K ₁₁ = 23.5/msμm ² in the mid-wavelength infrared frequencies is considered for the simulations. DFLC cell embedded with microstructures is operated between f _L = 10 KHz with dielectric anisotropy, Δϵ = 20 and f _H = 100 KHz with Δϵ = −2.0. Liquid crystal molecules orient parallel to the applied field (Z-direction) at low frequency, f _L and rotate perpendicular to the field (X-axis) at high frequency, f _H as shown in the Figure 1. We investigate the electromagnetic response of LC device operated with weak anchoring on both substrates and at the microstructures.

Incorporating the director orientations averaged in each layer along Z-axis n ̄ ( x , z ) as shown in the Figures 2C,F), we perform electromagnetic simulations based on FEM method using commercial software CST. Linearly polarized light is incident along Z-axis and incident polarization direction is along Y-axis. Scattering parameters (S-parameters) are defined by S ₁₁ = R and S 12 = T e i k 0 d , where “d” is the thickness of the metasurface, k ₀ is the wavenumber of the incident light in vacuum, R and T are the reflection and transmission coefficients.

Scattering response of cylinders for both the layer-wise averaged director angles ( n ̄ ( x , z ) ) and average director of the cell n ̄ (does not consider the geometry-induced director distortions) are shown in Figures 3A,B). Here n ̄ = z for “ON” state and n ̄ = x for “OFF” state. In the sub-wavelength regime exhibited above 30 THz for both the alignments, vertically aligned state (“ON” state) gives rise to multiple narrow bands as shown in Figure 3A, while the planar alignment (“OFF” state) exhibits a single broad transmission band in the range 30, −,44 THz. The transmission band in the “OFF” state is observed to have fano resonance type of characteristic with prominent peaks at selected frequencies. As observed from the Figures 3A,B), the shift in the response frequencies and the characteristics of the spectra for n ̄ and n ̄ ( x , z ) differ more significantly for the “ON” state compared to the “OFF” state. Scattered light further undergoes a gradual phase variation within the transmission band (29, −,37.5 THz for “ON” state and 30.5–38 THz for “OFF” state) as shown in the Figures 3C,D) and hence acts as tunable phase retarder within this broad frequency range. However abrupt shifts in frequencies are observed for “ON” state as shown in the Figure 3C). Interestingly variation in phase of the scattered light between n ̄ and n ̄ ( x , z ) is more prominent in the “OFF” state. While the n ̄ case shows a variation of 2π in transmission band in both “ON” and “OFF” states, a more realistic consideration of n ̄ ( x , z ) gives rise to lesser phase variation. At some of these resonance frequencies, complex hybrid and higher order modes are observed as can be seen from the near-field electric field profiles along XY- and YZ-planes in the Figure 3E). Significantly, “ON” state exhibits radiating hybrid modes at the frequencies indicated while “OFF” state gives rise to toroidal resonances (31 and 46.76 THz) and higher order modes (43.7 THz) with near-zero transmission in the propagation direction as shown in the Figure 3E).

Scattering parameters for cylindrical rings show similar behavior of multiple sharp responses in “ON” state and broad asymmetric transmission bands characteristic of fano resonance in “OFF” state as shown in the Figures 4A,B. As observed in the case of cylinders, the characteristics of the responses are quite similar for n ̄ and n ̄ ( x , z ) in the case of planar alignment. Phase variation in the case of “ON” state is sharper for cylindrical rings (at a single frequency) compared to cylinders while the “OFF” state exhibits a smooth variation of 2π for both n ̄ and n ̄ ( x , z ) cases in the frequency range 50, −, 56 THz as observed in Figures 4C,D. Figure 4D further shows that cylindrical rings in planar alignment exhibit efficient phase retardation compared to the solid cylinders. Near-field profiles of the cylindrical rings show presence of hybrid modes in both “ON” and “OFF” states. It is observed that the ring structures do not support non-radiating toroidal modes unlike cylinders. A detailed multipole analysis will help in further identifying the dominant resonant modes in all the above cases.

Far-field radiation (XZ plane) and angular distribution of electric and magnetic far-field profiles (XY plane) are plotted for cylinders and cylindrical rings incorporating n ̄ ( x , z ) at selected resonance frequencies in the Figures 5, 6, respectively. In the case of cylinders (Figure 5), far-field radiation at 31.1 THz in “ON” and, “OFF” states are complimentary to each other. This is due to the excitation of electric dipole mode in the “OFF” state in addition to magnetic dipole (observed in both the states) along Y-direction resulting in destructive interference along forward propagation. Hence the metasurface mediated DFLC cell acts as a polarization converter and switches from forward to backward propagation of scattered light at this frequency. Further one can also observe transition from magnetic dipole modes to hybrid modes (magnetic quadrupole + magnetic dipole in this case) and vice versa as observed at 43.7 and 46.76 THz, respectively.

Cylindrical rings exhibit more complex response spectra and near-field profiles compared to the cylinders which is also reflected in the far-field behaviour. For example, at one of the resonant frequencies, 49 THz the far-field radiation switches between backward to forward propagation at “ON” and “OFF” states. “ON” state exhibits both electric and magnetic dipoles while “OFF” state exhibits magnetic dipole + quadrupole hybrid mode. At 50.5 THz (Figure 6), “ON” state shows onset of magnetic quadrupole mode along with magneitc dipole at an angle 45° while the “OFF” state shows both electric dipole and magnetic dipole oriented at an angle 15° from X-axis giving rise to backward propagation. At 59.1 THz, the electric and magnetic dipole modes orient normal to each other for “ON” and “OFF” states hence the incident polarization vector shifts from Y-axis to X-axis as we switch the DFLC cell from vertical alignment (“ON”) to planar alignment (“OFF”) and vice versa. Far-field radiation at this frequency shows scattering along both forward and backward directions for “ON” state while the “OFF” state exhibits strong forward propagation. Cylindrical rings hence give rise to rich near- and far-field responses compared to cylinders in both “OFF” and “ON” states with versatile functionality as dynamic metasurfaces.

The response times of a typical liquid crystal under the influence of an aligning field can be estimated as a function of the applied voltage. The rise time of the LC molecules to orient along the aligning direction is given by [33,39]: τ r = γ d 2 K π 2 V V t h 2 − 1 , (2) where V _th is the threshold voltage, V is the voltage applied to drive the LC molecules, γ is the rotational viscosity of the LC material, d is the thickness of the cell and K is the elastic constant (bend constant K ₃₃ in the present case of VA cell). The relaxation time or the decay time, τ _d or τ ₀ of the LC molecules depends on the LC reorientations and hence on the elastic properties of the medium as τ d = γ d 2 K π 2 (3)

In DFLC since both the “ON” and “OFF” states are driven by high voltages the response times can be written in terms of the decay time,τ ₀ and voltage amplitudes as τ O N = τ 0 V l V t h l 2 − 1 (4) τ O F F = τ 0 V h V t h h 2 − 1 (5) where V _h and V _l are the corresponding driving voltages at f _H and f _L , respectively. V t h h and V t h l are the corresponding threshold voltages. Using the above equations, one can calculate the response times of the microstructure embedded DFLC cell as a function of V/V _th as shown in the Figure 7. It is observed from the figure that the rise time τ _ON ≤ 1 ms for V < 0.5V _th . As the applied voltage approaches V _th , τ r * diverges. The decay time calculated from the above equation show a similar behaviour as a function of V/V _th . Threshold voltage V _th can be approximated theoretically from the elastic properties of LC as given below: V t h = π K 33 ϵ 0 | Δ ϵ | (6)

From the above equation, threshold voltage for reorientations of LC molecules is given by V _th = 0.9 V at f _L (ON) and V _th = 8 V at f _H (OFF), respectively. From this we show that the response times of the LC orientations to the external driving voltages falls in the range of 1 ms and hence switching between electromagnetic responses at “ON” and “OFF” states as discussed in the earlier sections can be performed in ultrafast times. This opens up possibilities to design multifunctional devices that can be operated in ultrafast times, with some of the functionalities demonstrated in the present work with cylindrical and cylindrical ring microstructures in IR frequencies.