Fig. 1. (a) Schematic of the working principle of transmission modulation using insulator–metal phase transition in VO2. The transmission efficiency T from the structure changes as the optical index of refraction varies with the phase transition. The crystalline structures of the low-temperature insulator phase and the high-temperature metallic phase are shown on the right-hand side [28]. The green spheres correspond to the vanadium atoms, and the blue spheres correspond to the oxygen atoms. (b) Enhancing the transmission contrast ΔT through nanostructuring. The hypothetical enhanced performance (orange curve) as compared to the transmission through an 80 nm VO2 thin film on a silicon substrate (purple curve).

Fig. 2. (a) Schematic diagram of the metagrating studied in the parameter sweep. The incident wavelength is set to 1.55 μm, and the geometrical parameters include the grating height h, period Λ, filling ratio fr=wr/Λ, and polarization. (b) The maximally positive (green shaded) and negative (orange shaded) contrast ΔT over the period for different polarization conditions (blue for TM polarization, red for TE polarization). Two high-contrast regimes can be recognized and highlighted by the circles. The first regime has a positive contrast ΔT=0.63 with a deep-subwavelength periodicity (Λ≪λ), while the second regime has a negative contrast ΔT=−0.73 with a period close to the wavelength (Λ∼λ). For comparison, the best performance of the bare VO2 film is also shown with a constant value of 0.26 (black dashed line).

Fig. 3. Sweep of geometric parameters fr and h in a grating structure with periodicity Λ=50  nm, incident wavelength λ=1.55  μm, and TM incident polarization. The 2D plots display the transmission efficiency of the VO2 grating in the (a) insulating Ti and (b) metallic Tm phase of VO2. The high transmission ridges in Ti are highlighted using black dashed lines. The transmission contrast ΔT of the grating is shown in (c). The optimal geometry with maximum contrast is highlighted by the red star with fr=0.83 and h=370  nm.

Fig. 4. (a) Schematic of the investigated structure (top) and comparison with the geometry in the effective medium regime (bottom). (b) Electric field amplitude distributions of the corresponding subwavelength grating at 1.55 μm, in the insulator (i) and metallic (ii) VO2 states. (c) Reflectivity spectrum of the corresponding grating, where the amplitude |r| and phase arg{r} are displayed by red and blue curves, respectively. The solid line denotes the lossless case with VO2 material losses fully turned off, while the dashed line denotes the lossy case when the losses of the VO2 material are considered.

Fig. 5. (a) Diagram of the approximate zeroth-order effective medium theory (EMT). (b) Comparison between the predicted maximum ΔT using EMT (black solid curve), 1D grating with TM excitation (black dashed curve), unpatterned VO2 film (black solid curve), and topology optimization (solid blue curve). The comparison shows the numerical optimization is well bounded by the theoretical prediction. (c) Effective index of refraction calculated with the EMT using random geometrical parameters. The results are shown by the distributed points, where the top and bottom n−κ maps correspond to the insulator and metallic phase of VO2, respectively. The black edges correspond to the effective indices of the 1D grating geometries with TE or TM illumination.

Fig. 6. (a) Optimal geometries obtained from topology optimization using different thicknesses of the VO2 scattering structures. A common feature can be distinguished among all these geometries: the thin gap between adjacent structure elements is along the electric field direction. On the bottom, we report the geometry with the highest contrast ΔT=0.73, obtained for h=600  nm, Λ=50  nm, and incident wavelength λ=1.55  μm, which is well within the deep-subwavelength regime. (b) Calculated electric field profiles of the optimal structure (i), (ii) in its insulator state, and (iii), (iv) in its metallic state. Calculations (i) and (iii) show the side view of the structure (perspective along the x axis) with the cut plane located at y=0, while (ii) and (iv) show the bottom view (perspective along the z axis) with the cut plane located at z=0. The green contour marks the structure’s edges.

Fig. 7. Parameter sweep of geometric parameters fr and h in a grating structure with periodicity Λ=1300  nm, incident wavelength λ=1.55  μm, and TE incident polarization. The 2D plots display the transmission efficiency of the VO2 grating in the (a) insulating (Ti) and (b) metallic (Tm) phase. For Ti, the low transmission valleys are highlighted by the white dashed lines. The associated negative transmission contrast −ΔT is presented in (c). The red star highlights the optimal geometry with maximum negative transmission contrast for the parameters h=640  nm and fr=0.05.

Fig. 8. (a) Schematic diagram of the design having maximally negative contrast at λ=1.55  μm wavelength, with geometrical parameters Λ=1300  nm,fr=0.05, and h=640  nm. The structure is illuminated with TE polarization, i.e., an electric field parallel to the grating ridges. The calculated electric field amplitude and phase profiles of the corresponding structure in the insulator phase of VO2 are shown in (b) and in the metallic phase of VO2 in (c).

Fig. 9. (a), (b) Spectra and (c), (d) field patterns of the VO2 grating in the insulator state. For the realistic lossy case given in (a), (c), a transmission suppression dip is shown in (a) centered at the target wavelength 1.55 μm. In panel (c), the corresponding electric field distribution (shown by the background density plot) and also the flow of Poynting vectors (red arrows) are displayed. The material loss is turned off for (b) and (d). In (b), the broadband transmission suppression is centered at 1.45 μm. The corresponding field pattern and Poynting vector flow are displayed in panel (d).

Fig. 10. (a) Calculated electromagnetic behavior of a single rectangular rod. In particular, it shows the amplitude of the electric field, as indicated by the color bar on the right, together with the flow of the Poynting vectors, as indicated by the red arrows. (b) The spectrum of the retrieved effective magnetic permeability (μeff). A peak is presented in the imaginary part at the wavelength of interest 1.55 μm.