Fig. 1: High-order diffraction in focusing.
a Optical diffraction from a single zone that is sketched in a polar coordinate. b Simulated amplitude (normalized to the maximum) of light diffracted from the zones with different widths Δr. By changing Δr, the simulation is implemented by using Rayleigh-Sommerfeld diffraction with a fast Fourier transform17, where the pixel pitch of 0.02 λ × 0.02 λ is used to enhance the calculation accuracy. The zones have the same central position of (rn + rn-1)/2 = 75 λ for investigating the role of Δr. The diffraction orders of the zones are also labeled for better observation, where the dashed white curves (derived by using \(M=\left\lfloor \triangle R/\lambda \right\rfloor\)) denote the boundaries between two neighboring orders. For each Δr, the fast oscillations along the radial positions are not observed in current figure due to the large range in radial position. c On-axis (ρ = 0) intensity of diffracted light from axisymmetric zones with different widths Δr and the outermost radii rn (where the parameter \(\sin \alpha={r}_{n}/\sqrt{{f}^{2}+{r}_{n}^{2}}\) is used with f = 75 λ for better data display). The on-axis intensity of each order (labeled in different colors) has been normalized to the maximum of its own order. Two cases for the destructive (A with Δr = 4.69 λ and \(\sin \alpha\)=0.45) and constructive (B with Δr = 7.6 λ and\(\,\sin \alpha\) = 0.5) interferences have been exemplified to show its diffraction patterns (see Supplementary Fig. 1). The structural parameters of the 0th-order (dots) and 1st-order (triangles) FZPs are located at the region of constructive interference, implying diffraction-limited focusing similar to traditional objectives lenses. The destructive interference is indicated by white regions. d–e Simulated optical focusing via a standard 0th-order FZP (d) and high-order (e) FZP with binary phase. Optical paths from the nth radius to the focus are Rn = f + n λ/2 for standard FZP and Rn = f + n(2 M + 1)λ/2 for Mth-order FZP, which indicates their fundamental differences. f Analytical (gray bars) and simulated (blue-white-red bars) efficiencies of high-order FZPs with different NAs.
