Monoclinic nonlinear metasurfaces for resonant engineering of polarization states

2.1 Polarization transformation in linear transmission

Here we examine the manifestation of the chiral mode of choice in the context of polarization transformation for the linearly polarized input. We set the input field to be a monochromatic linearly polarized plane wave at frequency ω and wave vector k = − z ̂ k (we assume a e^−iωt time dependence):

where angle φ shows linear polarization orientation with respect to the x-axis, x ̂ , y ̂ , z ̂ are the Cartesian unit vectors. Based on Eq. (1), this implies the input Stokes parameters to be

We incrementally change polarization angle φ in the range [0, π] with a constant step Δφ, as the results are π-periodic.

Adjusting the polarization angle φ and the wavelength λ of the incident wave, we examine the polarization of the outgoing signal (Figure 3c). Near the chiral resonance, the circular polarization degree σ is strongly dependent on λ and φ. At certain λ and φ, the transmitted field approaches circular polarization, |σ| ≈ 1. Although the input polarization states (4) are evenly spaced, the output states are not. This is illustrated on the Poincarè sphere in Figure 3e. In particular, the output states form a circle on the sphere, indicated by red dots.

To validate our theoretical prediction regarding the polarization change, we measure all Stokes parameters of light transmitted through the metasurface taking into account Eq. (1) (see Supplemental Material for experimental details). The experimentally determined values of σ, as a function of the wavelength and the input polarization angle, are presented in Figure 3d. The output light exhibits elliptical polarization around the optical resonance. The behavior observed qualitatively agrees with the theoretical protection. However, the degree of circular polarization varies from −0.18 to 0.26 at 1,572 nm. The output polarization state is illustrated in Figure 3f by red dots, while the input polarization state is also depicted by blue dots. The experimental pattern is similar to the theoretical prediction. The difference between theoretical and experimental results is primarily attributed to fabrication imperfections and the broad range of incident light wave vectors in the experiment, which result in different mode excitations to the theoretical prediction. We provide theoretical and experimental results for wider range in Supplementary Materials.

2.2 Polarization engineering of the third harmonic generation

At higher intensities, the electron oscillations within a dielectric structure become anharmonic, which can be effectively described using the extended Lorentz model [40]. In its bulk crystalline form, silicon is a centrosymmetric material, making second harmonic generation (SHG) symmetry-forbidden (except surface effects [41]). Therefore, the third harmonic generation (THG) was examined to demonstrate the nonlinear behavior of the metasurface. The nonlinear polarization current responsible for THG can be expressed as P ( 3 ω ) = ε 0 χ ̂ ( 3 ) E ( ω ) E ( ω ) E ( ω ) , where ɛ₀ is the dielectric constant, E^(ω) is the electric field at the fundamental frequency, and χ ̂ ( 3 ) is the fourth-rank nonlinear susceptibility tensor. Silicon has space group m3m, which results in only 21 nonzero elements in χ ̂ ( 3 ) with only 4 independent [40]. While it is possible to find values of each component experimentally in some approximations, their values are usually of the same order of magnitude [42], [43]. For simplicity, we assume all non-zero components to be equal, e.g. as it is done in [44]. Moreover, in Supplementary Materials we show that even approximation of isotropic nonlinear response gives practically the same results. To model THG, we employ the undepleted pump approximation and simulate the process in COMSOL Multiphysics using a two-step approach. In this framework, the nonlinear polarization P^(3ω) is used as the initial condition for solving the higher harmonic wave equations.

The coupling strength of the incident field to the resonant mode strongly depends on the overlap integral between the two [28], [32], [34], [44]. This coupling shows a strong polarization dependence, providing different field distribution at the fundamental frequency E^(ω) for different input polarizations, and hence different P^(3ω) and third harmonic response.

We evaluate the nonlinear numerical results by calculating the THG emitted along the zeroth order for a linearly polarized pump and decomposing THG transmission signal into the right- and left-circular polarized components, I R ( 3 ω ) and I L ( 3 ω ) (Figure 4a). The wavelength-dependent THG intensities for right- and left-circular polarized components are shown in Figure 4c by the green and magenta lines, respectively. The results exhibit a pronounced resonant behavior revealing significant enhancement of the THG at the vicinity of the structure resonances. Remarkably, for the first resonance (1,555 nm) the right-circular polarized component dominates, while for the second resonance (1,634 nm) the left-circular polarized component becomes dominant. Next, we investigate the effect of varying the polarization angle of the pump at specific wavelengths and extract all polarization parameters for the THG. The simulated polarization states are represented on the Poincarè spheres in Figure 4b. Near the chiral resonances, the polarization trajectories for pump wavelengths of 1,555 nm and 1,634 nm exhibit high degrees of circular polarization, with |σ| values approaching 1. This indicates that highly circularly polarized THG is generated near these resonances. Unlike the linear case, the polarization points do not lie within a single plane and lack a clear pattern. This complexity likely results from the high density of resonant chiral states in the vicinity of 3ω (see Supplementary Material), where each mode shows a slightly different coupling coefficient – i.e. an overlap integral between the fundamental harmonic field and the high harmonic resonance – that varies significantly with the linear polarization angle with the resonance at the fundamental harmonic, as the field intensity is less homogeneous. In contrast, for wavelengths away from the resonances (e.g. 1,590 nm, as shown in Figure 4b), the output polarization closely mimics the input polarization, indicating the significance of the chiral resonance at the fundamental harmonic. This aspect requires further investigation and is not fully covered in the current paper.

Figure 4:

Nonlinear theory. (a) Third harmonic intensity output from a linearly polarized beam along x-axis. (b) Poincarè spheres which show simulated polarization states of the third harmonic output taken at different wavelengths for linear polarization input at different angles on the fundamental frequency. (c) The green and magenta lines are the intensity of the right and left circular polarization intensities of the third harmonic signal, respectively.

To test the theoretical predictions, we measure the THG from the metasurface (see Supplemental Material for more details). The laser wavelength was tuned from 1,500 to 1,730 nm at 5 nm intervals, while the polarization angle was varied from 0° to 180° with 10°. For each combination of these parameters, we record the THG spectra and extract the maximum values. Figure 5a shows the maximum THG value as a function of the pump wavelength and input polarization angle. We observe THG enhancement in both expected and unexpected spectral regions. The exact reason of such discrepancy between numerical simulations and experimental observation is rather unknown. However we speculate here on possible reasons:

1. imperfections in fabrications are much more noticeable on the scale of λ^(3ω)/n ≃ 125 nm (the typical wavelength of the TH signal inside metasurface material with refractive index n), specifically at the cylinders edges where the high-harmonic modes are mostly localized as they resemble whispery gallery modes;

2. difference in the refractive index dispersion used in simulations and dispersion of the real sample, as the change of the refractive index tend to shift the position of the resonances;

3. deviation of excitation shape from a plane wave, which was used in the theoretical calculations.

Figure 5:

Experimental nonlinear polarization properties. THG intensity (a) and degree of circular polarization (b) from the metasurface as a function of the pump wavelength and polarization angle. (c) Poincarè spheres illustrating polarization states of the output THG at 1,510 nm, 1,590 nm and 1,635 nm wavelengths for linear polarized input, varying linear polarization angle at the fundamental harmonic.

Numerical simulations of the THG intensity signal of the range shown in Figure 5a and typical mode profiles at 3ω frequencies are shown in Supplementary Materials.

To investigate the degree of circular polarization σ we extract this parameter from the THG data taking into account Eq. (1) – the dependence on the pump wavelength and input polarization angle is shown in Figure 5b. The results reveal complex dependencies with significant changes in the polarization state as the pump polarization angle. We provide a possible justification in the Supplementary Material. We further extract the polarization parameters for the resonant wavelengths 1,510 nm and 1,635 nm. These values are plotted on Poincarè spheres, showing σ ranges of −0.73 to 0.71 at the wavelength of 1,635 nm and −0.74 to 0.70 at 1,510 nm (Figure 5c). Like the theoretical results, these points do not lie in a single plane as we would expect from the experimental results, and reach higher values for circular polarization than those in the linear regime. In the non-resonant regime at 1,590 nm, while σ values are non-zero (−0.39 to 0.58), the low output intensity means other fabrication errors were likely to play a large role. The deviation between the experimental and theoretical results is attributed to the same factors as in the linear case. Additionally, the THG exhibits greater sensitivity to all possible imperfections and pump parameters compared to the linear regime. Considering the impact of the modes at the THG wavelength, it is challenging to fully replicate the simulated result in a real finite sample.