Diffraction-limited operation of micro-metalenses: fundamental bounds and designed rules for pixel integration

TL;DR

This work analyzes the diffraction-limited performance of micro-metalenses integrated with pixel arrays, identifying hard (device size) and soft (nanostructure period) limits that bound metasurface operation. A vectorial diffraction framework based on the Stratton-Chu formalism links focal behavior to the aperture size $a$, focal length $f$, and wavelength $\lambda$, revealing a Fresnel-number–dependent regime map and the onset of focal shift as $a$ becomes wavelength-scale. Experimental validation with GaN metalenses at $\lambda = 617$ nm confirms the predicted regimes and the impact of diffraction on miniature devices, highlighting when metasurfaces outperform simple apertures. The study provides design rules for integrating metasurfaces with pixel matrices in compact imaging systems and outlines regimes and sampling considerations essential for diffraction-limited performance.

Abstract

Metasurfaces provide a compact, flexible, and reliable solution for controlling the wavefront of light. In imaging systems, micro-lens arrays are integrated with pixel matrices to reduce optical crosstalk, enhance photon collection efficiency, and improve spatial resolution. However, as the aperture size of the photonic devices decreases, fundamental limitations associated with diffraction emerge. Here, we theoretically analyze and experimentally demonstrate that these constraints also affect the performance of small functionalized apertures, including metasurfaces and metalenses, emphasizing the increasing impact of diffraction at small pixel sizes. Despite their design versatility, our findings reveal the necessity of accounting for fundamental diffraction properties to optimize the performance of miniature optical metasurfaces.

Figures (4)

• Figure 1: (a) Artistic representation of the integration of a metalens array with a matrix of pixels (b) Numerical simulation results summarizing the variation of the focal distance of metalenses designed with a fixed numerical aperture of $\text{NA}=0.2$ as a function of their diameters. The red curve represents the effective focal distance, and the black dotted line represents the designed focal length. The red area represents the effective full width at half maximum (FWHM) of the spot in the $z$-direction, as indicated in the insert. Finally, the blue curve represents the focal shift error $\Delta f/f = \left\lvert f_{eff}-f\right\rvert/f$ as a function of the metalens diameter. The simulations were carried for a wavelength of $617nm$.
• Figure 2: (a) Schematic of a circular transmitting aperture in an opaque screen capable of imparting an additional transmission phase delay function $\phi(\bm{r}^\prime)$ on the incoming light, as illustrated with a red overlaying profile. (b) Schematic of the diffraction pattern for a plain aperture, $\varphi(\bm{r})=0$. The vertical black dashed line represents the focal plane of the aperture, approximately located at $a^2/\lambda$. (c) and (d) show the normalized intensity (bottom panels) along the $z$-axis for two metalenses with fixed focal distance of $f=6a$ and with $ak =510$ (left panel) and $ak=51$ (right panel), respectively. Top panels show the focusing function $f(x) = \mathop{\mathrm{sinc}}\nolimits^2\left[ \frac{ak}{4}\left( \frac{1}{x}-\frac{a}{f} \right) \right]$ in black with the left ordinate and intensity decay $g(x) = \frac{(ak)^2}{4} \frac{1}{x}$ in red with the right red ordinate. For $ak = 510$, there is no focal shift, while for $ak = 51$, a focal shift occurs.
• Figure 3: (a) Fresnel number $N$ as a function of the normalized aperture size $a/\lambda$ and the numerical aperture. The blue line represents $N=1$, the green colors $N>1$, and the orange colors $N<1$. The dashed black lines represent power of $10$. (b) Number of elements per Fresnel zone $N_m$ as a function of the period of the metasurface and the numerical aperture. The black dot represents the designed metalenses fabricated and characterized with $p = 312.5nm$ and $a = 10µm$. In both figures the wavelength is taken equal to 617nm.
• Figure 4: (a) Scanning electron micrographs of the GaN metalenses with focal length (A) $f=8µm$, (B) $f=500µm$ and (C-D) $f=3000µm$. (D) The scale bar represents $2µm$ for (A-C) and $1µm$ for (D). (b) Evolution of the effective Fresnel number, $N_{\text{eff}} = a^2/\lambda f_{\text{eff}}$ and expected Fresnel number $N = a^2/\lambda f$. The red curve corresponds to the simulation results. (c) Comparison between numerical (bot panels) and experimental (top panels) intensities for metalenses of diameter $10µm$ and focal length $1000µm$ (left panels), $100µm$ (middle panels) and $10µm$ (right panels). The numerical simulations have been done using numerical integration in Eq. \ref{['eq:equivTh']}.