Theory and Fundamental Limit of Quasiachromatic Metalens by Phase Delay Extension

The periodic extension of phase difference is commonly applied in device design to obtain phase compensation beyond the system’s original phase modulation capabilities. Based on this extension approach, we propose the application of quasiphase delay matching to extend the range of dispersion compensation for meta-atoms with limited height. Our theory expands the limit of frequency bandwidth coverage and relaxes the constraints of aperture, NA, and bandwidth for metalenses. By applying the uncertainty principle, we explain the fundamental limit of this achromatic bandwidth and obtain the achromatic spectrum using perturbation analysis. To demonstrate the effectiveness of this extended limit, we simulate a quasiachromatic metalens with a diameter of 2 mm and a NA of 0.55 in the range of 400–1500 nm. Our findings provide a novel theory for correcting chromatic aberration in large-diameter ultrawide bandwidth devices.

Figure 1

(a) Phase extension with $2\pi$ interval between each pair of dashed lines. The stepped black lines represent the phase fitting at different frequencies. (b) PD extension. Blue and green lines represent $m=1$, 2. The circles represent the phase extension points supported by the meta-atoms. The orange dashed curves represent PD curves at the entire frequency domain. (c) Schematic of achromatic focusing of PD extension. The upper part of QAML demonstrates the wave distribution at focusing time, and the bottom part at incident time. The solid color lines represent light incident at $T_0$, while the dashed lines represent light incident at extended PDs.

Figure 2

(a) Achromatic spectrum of QAML and BAML bandwidth limit. The colored dashed line represents the ideal achromatic amplitude under flat spectrum input without nonlinear dispersion. (b) Comparison of bandwidth of QAML and BAML with their limits, where the QAML total sub-bandwidth exceeds the BAML bandwidth limit when $N_M=20$. The fundamental bandwidth limit of sub-band $\delta {\omega }_{\mathrm{R}}$ is also dependent to $N$. (c) Nonlinear dispersion analysis. The red region determined by the Raleigh criterion represents linear dispersion region.

Figure 3

(a) PSFs of QAML at extended frequencies. (b),(c) Focal lengths and focusing efficiencies of PSF at all frequencies. The star points represent deviation frequencies, and the focusing efficiency is obtained from the energy within 3 times the Airy disk radius. (d) Comparisons of published BAML database [12] and fundamental bandwidth limit of BAMLs and QAMLs. (e) Simulated normalized pseudoimaging results and MTFs under solar spectrum, LED, and optical frequency comb incident, respectively.