Fig. 1. Schematic diagram of the phase profiles of broadband achromatic metalenses. (a) The phase profile of a generally designed achromatic metalens. The gray dashed line is the theoretically calculated phase profile of focal length f. Due to the phase shift Δφ of the selected meta-atoms determined using a periodic boundary condition, the actual phase profile of the metalens becomes a brown solid line, and the corresponding focal length changes to f'. For other wavelengths in the bandwidth range, the focal lengths have similar changes, so that the focal lengths in the entire bandwidth range are not exactly the same, and the purpose of the wide bandwidth cannot be achieved. (b) The phase profile of a broadband achromatic metalens using APPCM. For a certain wavelength λ, the green solid line is the phase profile with focal length f₁ designed in the Z1 zone (inner), the blue solid line is the phase profile with focal length f₂ designed in the Z2 zone (outer), and the red solid line is the overall phase profile corresponding to the focal length f, which is between f₁ and f₂. The combination of both phase profiles and the compensation of the phase shifts caused by both the zones make the overall focus lengths stay almost the same for all wavelengths within the bandwidth.

Fig. 2. (a) Schematic of an achromatic metalens composed of cylindrical Si₃N₄ ride meta-atoms. The height of the hollow nano-cylinder is 1500 nm, and the period of the meta-atoms (P) is 450 nm. (b) Wavenumber dependences of transmission (blue) and phase (red) for the meta-atom. The high transmission and a nearly linear phase as a function of the wavenumber are achieved in the entire visible region. The inset shows the geometry of the hollow nano-cylinders, where the outer and inner radii of the ring are 170 nm and 70 nm, respectively.

Fig. 3. (a) Simulation of the phase manipulation of Si₃N₄ nano-cylinders. The light blue solid dots represent the phases imparted on the nano-cylinders with different inner and outer radii at the maximum and minimum wavelengths. The black and green stars are the theoretically calculated phases for the Z1 zone (f₁ = 200 µm) and the Z2 zone (f₂ = 250 µm), respectively. It should be noted that some black stars and green stars overlap due to different diameters and FLs for Z1 and Z2. The first black star on the upper right and the last green star on the lower left represent the first unit and the last unit, respectively. All units are arranged in order from top to bottom to be matched. (b) Phase profiles of the broadband achromatic metalens M1 designed using the APPCM approach. The phase manipulation using the chosen nano-cylinders (red) is basically following the theoretical calculations (blue).

Fig. 4. Normalized intensity distribution of the broadband achromatic metalens M1. (a) Simulated normalized intensity profiles in the plane x = 0 (y–z plane) around the focal point of metalens M1 at different wavelengths. The gray dashed line is the average (214 µm) of the focal lengths for all wavelengths. (b) Intensity profiles in the focal plane (x–y plane) of M1 at different wavelengths. (c) Longitudinal cross sections of the corresponding focal spots in (b).

Fig. 5. Performances of broadband achromatic metalenses M1 and M2. (a) Focal length as a function of wavelength for metalenses M1 and M2. The round-dot line and the diamond-dot line represent the focal lengths of M1 and M2, respectively. The red and blue lines are the focal lengths of the entire metalens, whereas the gray line is the focal length of the inner zone. The green and yellow shadow areas demonstrate the DOF of M1 and M2 at different wavelengths, respectively. (b) Simulated focal lengths of M1 for X, Y polarizations (X-LP, Y-LP), right-handed circular polarization (RCP), and right-handed elliptical polarization (REP). (c) Simulated FWHMs at different wavelengths. The red round and blue diamond dots are the FWHMs of M1 and M2, respectively. The green and yellow dotted lines represent the theoretical FWHMs of metalenses M1 and M2. (d) The red round and blue diamond dots give the wavelength dependence of the focusing efficiency of metalenses M1 and M2, respectively.