Supporting Information: Achromatic Metalens over 60 nm Bandwidth in the Visible and Metalens with Reverse Chromatic Dispersion M. Khorasaninejad 1*, Z. Shi 2*, A. Y. Zhu 1, W. T. Chen 1, V. Sanjeev 1,3, A. Zaidi 1 and F. Capasso 1 1 Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA 2 Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA 3 University of Waterloo, Waterloo, ON, N2L 3G1, Canada *These authors contributed equally to this work. Emails: Khorasaninejad@seas.harvard.edu, capasso@seas.harvard.edu Note 1: Guided Mode Resonance Figure S1a shows the schematic of a TiO 2 array on a metallic mirror with a thin layer of silicon dioxide (spacer) in-between. Normally incident light (entering from air) can take two possible optical paths. In the first optical path (Figure S1a), light propagates along z-axis via nanopillars and the spacer layer and then it is reflected back by the metallic mirror. In general, light can bounce several times before is coupled out to air. This can be seen as a Fabry-Perot cavity. In addition, this structure can support waveguide modes with propagation direction in x-y plane. In the latter, the array of TiO 2 nanopillars serves as the core of the waveguide with air and spacer as the cladding layers. This provides the second optical path (Figure S1b). In order to couple into these guided modes, the phase matching condition must be fulfilled. This can be done by adjusting the center-to-center distance of the nanopillars (U) such that (m : integer) matches the propagation constant of the corresponding guided modes (β). Therefore, U should be larger than the effective wavelength inside the waveguide core (TiO 2 in our case). Additionally, it should be smaller than free space wavelength to avoid higher diffraction orders into air. Due to reciprocity, these modes are intrinsically leaky and can be coupled out into the air. The final amplitude and phase of reflected light is the superposition of these optical paths. To study the guided mode resonance supported by a TiO 2 array, we consider a twodimensional case (Figure S2a) that is periodic along x-axis and infinite along y-axis 1
(TiO 2 nanoridge). First, we assume the width of the TiO 2 nanoridge is equal to U (see Figure S2b, filling factor f=1). In this case, there is no coupling to guided modes because the phase matching condition cannot be satisfied. Therefore, light only can take the first optical path. The intensity and phase of reflected light as a function of wavelength are shown in Figure S3a and Figure S3b, respectively. As expected, Fabry-Perot resonances result in several dips in reflection. Next, by reducing the width, thus introducing periodicity, we can excite guided mode resonances (f < 1). This is evident in the reflection curve of Figure S3c, where several sharp resonances appear at the filling factor of 0.916 (W=0.916U, W=440 nm, U=480 nm). These resonances also have large effects on the phase compared to the previous case. They not only expand the phase coverage, but also result in anomalous dispersion, where the first derivative of the phase with respect to wavelength changes sign around the resonance wavelength. To verify the guided mode resonance nature of these resonances, we calculate the resonance frequencies for different filling factors using finite difference time domain simulation (Figure S4) considering periodic boundary condition along the x-axis and perfectly matched layer along the z-axis. At the same time, we calculated the dispersion relation for the case of f=1 using Transfer Matrix method (Figure S5). As expected, the guided mode resonance frequencies in the weak perturbation limit (filling factor approaches unity) approach those of the waveguide modes with propagation constant β= (calculated by Transfer Matrix method, see Figure S4&S5). This confirms the assumption that the resonances seen in simulation come from the excitation of guided modes. For simplicity, we consider a 2D case. The extension to 3D is straightforward, and the physical origin of the resonances is essentially the same. In the 3D case, the structure becomes a photonic crystal slab on top of a mirror. Since the Brillouin zone is expanded (Figure S6) there is more than one direction that has periodicity ( X, X, M). Therefore, coupling into guided modes can happen in all three directions, giving rise to more complex resonance spectrum. 2
Figure S1. Schematic of structure consisting of titanium dioxide (TiO 2 ) nanopillar with width W, center to center distance U=480 nm, and height H=600 nm, on a substrate. The substrate is an aluminumcoated fused silica with a thin film of silicon dioxide on top. Aluminum and silicon dioxide have thicknesses M=110 nm and S=180 nm, respectively. (a). Illustration of the first optical path, Fabry- Perot. (b). Illustration of the second optical path, guided modes. Figure S2. Schematic of 2D case. (a) array of titanium dioxide (TiO 2 ) nanoridges with width W, center to center distance U=480 nm, and height H=600 nm, on a substrate. The substrate is an aluminumcoated fused silica with a thin film of silicon dioxide on top. Aluminum and silicon dioxide have thicknesses M=110 nm and S=180 nm, respectively. (b) TiO 2 slab on the substrate, f=1. For both cases, the structures are infinite along y-axis. 3
(a) (c) (b) (d) Figure S3. (a)-(b) Computed intensity (a) and phase (b) of reflected light (see Figure S2b) as a function of wavelength for f=1. (c)-(d) Computed intensity (c) and phase (d) of reflected light (see Figure S2a) as a function of wavelength for f=0.916. For all simulations, polarization of incident light is along y- axis. Figure S4. Blue circle: frequency of guided mode resonance at different filling factors calculated by FDTD method. Red cross: frequency of the waveguide modes corresponding to the propagation constant (, m : integer) for filling factor f=1. This is calculated by Transfer Matrix method. 4
Figure S5. Dispersion relation of the waveguide modes for the case of f=1. Crosses show frequencies of the waveguide modes corresponding to the propagation constant ( =m 2π/U, m : integer), and they are also shown in Figure S4. X M X Figure S6. Brillouin zone for the 3D case. 5
(a) (b) (c) (d) (e) (f) Figure S7. Required phase (solid line) and realized phase (circles) at six wavelengths (a) 490 nm, (b) 505 nm, (c) 520 nm, (d) 535 nm, (e) 550 nm, and (f) 565 nm. It is notable that the reference phase (the required phase at the center of the achromatic metalens (x=y=0)) is different at each design wavelength as discussed in the main manuscript. Figure S8. Simulated intensity profiles of the reflected beam by the achromatic metalens in the xz-plane at several wavelengths. Here, we consider a systematic fabrication error where all nanopillar widths are 10 nm larger than the designed values. This results in an enhancement of secondary intensity peaks compared to those in Figure 4b of the manuscript. This effect is clearer at =490 nm where the previous secondary peak in the Figure 4b become the primary peak here. 6
Focusing efficiency(%) Figure S9. Measured focusing efficiency of the achromatic metalens as a function of wavelength. 7