Supporting Information

Transcription

1 Supporting Information Devlin et al /pnas Optical Characterization We deposit blanket TiO films via ALD onto silicon substrates to prepare samples for spectroscopic ellipsometry (SE) measurements. When more than one material is present for SE measurements, a model must be developed to extract the complex refractive index, ~nðωþ = nðωþ + ikðωþ, of a specific layer. In our case, we use a standard model for the substrate, and we apply the TL model for amorphous materials to the TiO film. The TL model that we use to extract the optical constants of our ALD TiO is a combination of the normal quantum mechanical Lorentz oscillator and the model Tauc derived for the imaginary part of the dielectric constant for amorphous materials above the bandgap. Tauc derived this model assuming a set of N noninteracting oscillators per unit volume and arrived at the following expression for the imaginary dielectric constant: «= A T E Eg E, [S1] where A T is the amplitude of the oscillator and E g is the transition energy. For the Lorentz oscillator, the imaginary part of the dielectric function is given by A L E 0 CE «= nk = E E0, [S] + C E with E 0 being the resonanto energy of the oscillator and C accounting for broadening. Jellison and Modine (1) combined Eq. S1 and Eq. S to account for shortcomings present in each model, leading to the TL model for the dielectric constant (1) 8 >< ACE 0 E Eg «= E E E > E g 0 + C E, [S3] >: 0 E E g where A is the product of A T and A L ; we have defined the other fitting parameters above. The real part of the dielectric function is then obtained through Kramers Kronig integration. The values of the four fitting parameters as well as the thickness of the film are shown in Table S1. Fig. S1 shows the raw ellipsometry data Ψ and Δ, and the corresponding generated data based on the model discussed above. Structural Characterization As stated in in the main text (discussion surrounding Fig. 1B), the measured surface roughness of the TiO films is on the order of surface roughness of the underlying substrate. Fig. S shows an AFM scan of the fused silica substrates used throughout this study. From the image, we extract an RMS roughness of nm. This value is comparable to the measured surface roughness of a fused silica substrate with a film deposited on top (0.738 pm). To determine the atomic structure of the ALD TiO, we use X-ray diffraction (D8 Discover; Bruker). As can be seen from Fig. S3, there are no signs of any diffraction peaks from TiO,even over the large-angle scan. We measure this diffraction spectrum for a TiO film on a 1-inch-diameter fused silica substrate and align the substrate to the center of the X-ray beam via a laser alignment mark. The absence of diffraction peaks indicates that the TiO films that we deposit are amorphous. In contrast, if the film was polycrystalline, the different polymorphs would generate diffraction peaks at 7.35 for rutile or 5 for anatase, for example. Geometric Phase We use PB ( 4) phase optical elements to implement the holograms, as first detailed by Bomzon et al. (1). In our case, the individual elements are TiO nanofins that act as discrete uniaxial crystals they possess a structural birefringence that leads to a fast and slow optical axis, which introduces a phase difference between orthogonal components of the incident electrical field. It is then a spatially varying rotation of the fast axis of each nanofin that gives rise to the geometric phase accumulation, as detailed by Pancharatnam (6) and Berry (7). Mathematically, in Jones calculus, a waveplate with spatially varying fast axis, in the basis of left and right circularly polarized light (LCP and RCP, respectively), can be represented by the matrix (5) φ Tðx, yþ = cos 1 0 φ i sin exp½iθðx, yþš, exp½iθðx, yþš 0 [S4] where φ is the retardance of the waveplate and θðx, yþ represents angle of rotation of a waveplate at position ðx, yþ. The matrix above allows one to take a beam of arbitrary input polarization, E i, and find the output state φ φ E 0 = TE i = cos E i i sin ½hE i jriexpð iθðx, yþþjli [S5] + he i jriexpð iθðx, yþþjliš, where R and L represent the left and right circularly polarized basis vectors and he i jr, Li is the projection of the input polarization onto the RCP and LCP basis, respectively. As shown by Bomzon et al. (1), a particular case of interest occurs when the polarization of the input beam is RCP or LCP and the retardation, φ, is π. In this particular case, the efficiency of the system goes to unity, and the output state for an input of RCP light becomes E o = expð iθðx, yþþjli, [S6] which shows that the output polarization is the inverse of the input polarization and the output of the beam has acquired a phase of θðx, yþ. By the symmetry of the half waveplate, the angle θðx, yþ can only vary from 0 to π, but the additional geometric of means that, by locally rotating the TiO nanofins, we can achieve full π-phase coverage. Metahologram Design, Simulation, and Measurement To create the holographic images shown in Fig. 4, we first take a binary image and produce a phase map via the Gerchberg Saxton algorithm (Fig. S4). We then run simulations (3D FDTD; Lumerical), using the measured TiO optical data shown in Fig. 1A. At a fixed height of 600 nm, we optimize the length and widths of the nanofins to provide the π-phase difference between two orthogonal components of the electric field (E x and E z ), as required for maximum efficiency. As can be seen from Fig. S5, at the design wavelengths of 480, 53, and 660 nm, the TiO nanofins provide a π-phase delay between the x and z components of the electric field and thus act as a half waveplate. 1of7

2 Although each individual nanofin must introduce a π-phase shift to have a maximum conversion efficiency (ratio of power in the output circular polarization and the power in the input circular polarization with opposite handedness), the total efficiency of the devices also depends on the transmission of each of the nanofins (5, 6). Fig. S6 shows the simulated transmission spectra of the TiO nanofins used for design wavelengths of 480, 53, and 660 nm. The simulations were run with a source polarized parallel to the long or the short axis, and the simulation setup is identical to that used in Fig. S5. The transmission remains relatively high for most designs throughout the entire visible. However, the design for 660 nm has relatively low transmission at shorter wavelengths; this is reflected in the measured efficiency and could be amended by allowing the optimization algorithm of the structures to search more exhaustively for a structure with higher transmission. Additionally, one could include antireflective coatings, taper the nanofins, or explore using overlapping electric and magnetic resonances (Huygens metasurfaces) to maximize the efficiency of all devices; this is beyond the scope of the current work. We collect the holographic images shown in Fig. 4 using the schematic shown in Fig. S7. A supercontinuum laser provides access to wavelengths from 470 nm to 800 nm and is sent through a collimator, linear polarizer, and quarter waveplate to inject circularly polarized light, as required by the PB phase (see discussion in Geometric Phase). The circularly polarized light is then incident on the μm metahologram contained on the sample and is converted to a holographic image with the opposite handedness of the input light. The light that passes through the sample is then sent through a 100 objective with a numerical aperture of 0.9. Because the spot of the collimated beam is larger than the area of the hologram, as well as to filter out any light that passes through the hologram but does not get converted (due to the TiO nanofin not working as a perfect half waveplate, for example), we place a filter in cross-polarization with respect to the input light after the objective. In the case of efficiency measurements, the intensity is then measured using a power meter. For the case of collecting the holographic images, we pass the light through a Bertrand lens to magnify the hologram. Simulations for General TiO Metasurface The process and TiO material properties demonstrated in Fig. 1A are not limited to metasurfaces implemented using PB phase. To show that we can produce different types of metasurfaces, e.g., those shown by Arbabi et al. (5), we simulate structures with the collected n and k data of our TiO and nanostructure dimensions achievable with our process. The simulation results are shown in Fig. S8. As we vary the diameter of a TiO pillar, fixed at a height of 600 nm, we can produce phase differences from 0 to π without using geometric phase. Fig. S1. Raw ellipsometry data used to determine the TiO optical constants, Ψ and Δ, for the TiO film on a silicon substrate as a function of wavelength. The green squares and circles are the values of Δ for angles of 55 and 75, respectively. The red squares and circles are the Ψ values for angles of 55 and 75, respectively. The black lines are generated data from the model described in Optical Characterization. of7

3 Fig. S. Atomic force microscope image of bare glass substrate with a root-mean-squared roughness of nm. Fig. S3. X-ray diffraction of ALD TiO. There are no observable diffraction peaks from any TiO polymorphs (red line). The peaks that appear in the scan result from the X-ray diffraction stage as can be seen from comparing the scan with (red) and without (blue) the sample. 3of7

4 Fig. S4. Phase map of Harvard logo used to generate holograms for design wavelength of 480 nm. Inset shows a pixel phase distribution. This phase information was translated to pillar rotations. 4of7

5 Fig. S5. Simulated electric field profiles at design wavelength. Real part of the x component (Left) and z component (Right) of the electric field at design wavelengths (A) 480 nm, (B) 53 nm, and (C) 660 nm. All fields are shown through a cross-section of the nanofin width, and the nanofin is highlighted in each panel with a black box. One can see that the x and z components of the electric fields exiting the pillar, at each wavelength, are out of phase by π radians, as required for PB phase. The TiO pillars are simulated on a glass substrate that occupies the half space below y = 0, and the wave propagates in the +y direction. 5of7

6 Fig. S6. Simulated transmission spectra for TiO nanofins on a periodic lattice at design wavelengths of (A) 480 nm, (B) 53 nm, and (C) 660 nm. In each panel, the solid (dashed) line corresponds to an incident plane wave source polarized along the long (short) axis of the rectangular nanofins. Bertrand Lens LP /4 /4 LP Screen SuperK 100x Fiber Coupled Collimator Sample /4=Quarter waveplate LP:Liner polarizer Fig. S7. Schematic of measurement setup for collecting holographic images. LP, linear polarizer; λ/4, quarter waveplate; SuperK, Supercontinuum laser. 6of7

7 1 Ex 0.8 Height (mm) D = 70 nm D = 90 nm D = 15 nm D = 15 nm D = 5 nm D = 75 nm - Fig. S8. Simulation of full π-phase coverage using TiO nanopillars with varied diameters. The white dashed lines show the placement of the TiO nanopillars, and the diameter of each pillar is listed below. Around each pillar, a 35-nm cross-section of the x component of the electric field is included. The black dashed line is set to 600 nm, the height of the nanofins that we presented in the main text. Table S1. Parameter Fitting parameters for TiO film Value Thickness, nm ± A, ev 4.4 ± 18.4 E 0, ev ± C, ev ± E g, ev ± of7