Computational inverse design for ultra-compact single-piece metalenses free of chromatic and angular aberration

Modern augmented reality/virtual reality (AR/VR) applications demand increasingly sophisticated optical components with ultra-compact form-factors that can deliver the same level of performance as their conventional bulky counterparts. Metalenses and multilevel diffractive optics^1–7 offer an innovative solution by replacing the bulky refractive components with much more compact interfaces, but they have not fundamentally altered the familiar multilens stacking strategy that uses ray optics to simultaneously minimize chromatic and off-axis aberrations [Fig. 1(a)]—a configuration that still consumes a large part of the device volume.^8,9 We propose comprehensive nanophotonic solutions for miniaturization of optics, which reduce the role of ray mechanics and free-space propagation, by exploiting the much richer wave interactions occurring in nanostructured photonic media. Specifically, we present single- piece nanophotonic plan-achromats [Fig. 1(b)] for achromatic, aplanatic, and curvature-free focusing, that is, realizing diffraction-limited focal spots [Strehl ratio (SR) >0.8 and absolute focusing efficiency (AE) of up to 65%] at precisely prescribed positions on a flat sensor over finite spectral and angular bandwidths. Our proposed solutions also allow for shortening the lens-to-sensor distance [Fig. 1(c)] by tailoring the output angles while maintaining the plan-achromaticity. Our primary goal in this work is to demonstrate the vast number of functionalities that can be incorporated into a small thickness (⁠$∼10λ$⁠, where λ is the operating wavelength) of low-index polymer structures by exploiting photonic inverse design,^10–12 enabling over four orders of magnitude (⁠$>104$⁠) reduction in device thickness compared to traditional multilens systems. Rather than constraining our techniques to any specific fabrication method, our theoretical results reveal the untapped potential of single-piece 3D metaoptics, motivating future work to accelerate the advancement of nanoscale 3D fabrication technologies^13–17 and computational methods that will incorporate the evolving constraints of such technologies^18–20 into 3D nanophotonic inverse design.

Previous work on metalens optics has been primarily devoted to achieving achromatic focusing at the normal incidence^3,21,22 or correcting off-axis aberrations at a single frequency.^6,7 In particular, a number of inverse design techniques have been explored to maximize metalens efficiencies over either spectral or angular bandwidths; such techniques involve combining locally periodic approximations, domain decomposition methods, or even full-wave Maxwell solvers with large-scale optimization algorithms^10,12 to discover multifunctional metalens geometries.^{14,15,23–28} However, achieving single-piece plan-achromaticity (i.e., aberration-free focusing over simultaneous spectral and angular bandwidths) remains an open problem. While plan-achromaticity may be achieved by a multiplet metalens configuration,²⁹ such an approach still requires considerable propagation distance between individual lenses, which presents a significant hurdle to further reducing the device footprint. In contrast, our computational results indicate that it is possible to design single-piece unibody plan-achromats, enabled by utilizing 3D nanophotonics, thus circumventing the necessity of multiplet ray-optics designs.

In this work, we employ a topology-optimization³⁰ based inverse-design tool to design the metalenses. Topology optimization (TO) is a computational technique for inverse design that can handle extensive design spaces [$>109$ design variables (DVs)³¹] considering the dielectric permittivity at every spatial point as a DV.^10–12 In contrast to heuristic search routines such as genetic algorithms³² or particle-swarm methods,³³ TO employs gradient-based optimization techniques to explore hundreds to billions of continuous DVs. Such a capability is made possible by a rapid computation of gradients (with respect to all the DVs) via adjoint methods,³⁴ which require just one additional solution of Maxwell's equations^10,12 in each design iteration. In fact, these techniques have been gaining traction lately in the field of photonic integrated circuits, especially for the design of compact modal multiplexers and converters.³⁵ More recently, TO-based inverse design methods have been extended to metasurfaces.^{15,23,24,26,27}

Here, we employ TO to design a single-piece nanophotonic plan-achromat, which focuses incoming plane waves at multiple frequencies and multiple angles. The focusing quality is measured by the Strehl ratio (SR); the SR above 0.8 is typically accepted as a gold standard for aberration-free imaging.² The SR is defined as

where $Ppeak$ is the peak intensity at the focal point and $Pairy$ is the peak intensity of a diffraction-limited ideal airy disk normalized by the same transmitted power as that above the metalens. In particular, to design a plan-achromat, we maximize the minimum of SRs for multiple frequencies and angles ${(ωi,θi):i=1,…,N}$⁠; such a problem can be equivalently formulated in an epigraph form to obtain a differentiable optimization problem,³⁶ using an extra variable t,

In contrast to typical TO, where the relative permittivity $ε(r)$ is varied at each pixel, we consider varying the thickness profile of each layer¹⁵ with an aim to leverage the capabilities of 3D printing^15,37 and thermal scanning probe lithography³⁸ for fabrication and characterization in future works.

In Fig. 2, we present a proof-of-concept 2D cylindrical lens design (invariant along y axis), which can focus 10 frequencies and 10 angles (N = 100) within a spectral and an angular bandwidth of 23% and 60°, respectively. The lens is $50λ$ wide and has a numerical aperture (NA) of 0.24; it consists of 20 layers of 3D-printable polymer (refractive index, $∼1.5$⁠), corresponding to a total thickness of $12λ$⁠. Remarkably, TO discovers a design with a fairly uniform $SR≈0.89$⁠, achieving aberration-free diffraction-limited focusing for all the design frequencies and angles; the uniformity is a typical by-product of the epigraph formulation.³⁶ In between the frequencies and angles targeted in the design process, the mean SR is >0.75 (Fig. 2, top-right inset). In addition to the SR, it is also instructive to compute the absolute focusing efficiency (AE), defined as the fraction of transmitted power within three full-widths at half-maximum (FWHM) around the focal peak divided by the total incident power. Figure 2 shows simulated AEs whose average is 55%; while the AE of our design may fall short of the efficiencies typically seen in bulky refractive lenses, it is better than or comparable to any previous achromatic metalens design,^3,28 despite the multiple scatterings and complex interactions occurring within a multilayer geometry; further improvements in AE are likely possible if we also consider transmitted power as an additional constraint during optimization. The high SR implies that the 45% non-focused light is mainly lost to backscattering. Next, we demonstrate that our technique is not limited to small NA lenses; indeed, it can be straightforwardly scaled to higher NAs and larger lens widths. Figure 3 shows a wider and thicker design (⁠$125λ$ wide and $24λ$ thick) with a substantially higher NA of 0.7, achieving $SR>0.85$ for four frequencies and four angles with an average efficiency of 25%.

Having shown proof-of-concept 2D cylindrical lens designs, we now demonstrate that the proposed optimization framework can be used to design 3D plan-achromats. Designing a large-area fully freeform 3D plan-achromat would, however, demand much more intensive computational resources than were available to us in this work. A way to dramatically reduce the computational complexity is to exploit the axial symmetry (axisymmetry) of focusing.¹⁵ In other words, we restrict our 3D designs to be rotationally symmetric around a central axis. Doing this allows us to describe space using polar coordinates $(r,ϕ,z)$ and exploit the $ϕ$-invariance $ε(r,ϕ,z)=ε(r,z)$ (Fig. 4 top). This allows for the reduction of the 3D Maxwell equation to a set of independent 2D equations in (r, z), each characterized by an integer angular momentum number m,³⁹ 

where

The total electric field E is, then, given by the coherent sum of the individual fields times the exponential azimuthal factor (⁠$Emeimϕ$⁠) over all m's.

It is important to recognize that, in contrast to the axisymmetric material distribution ε, the incident current J may not necessarily obey axisymmetry. For example, linearly polarized normally incident plane waves may be decomposed into two counter-rotating circularly polarized waves (⁠$m=±1$⁠) both of which do obey axisymmetry. In contrast, any plane wave impinging on the lens at an oblique angle θ must be decomposed into an infinite number of axisymmetric components (with different m's), the sum of which may be truncated at a finite number of terms, depending on the desired accuracy of the approximation; this decomposition (known as the Jacobi–Anger expansion) is given by

where the integrals P and Q can be evaluated by Bessel's functions. Therefore, the total electric field response of an axisymmetric lens under off-axis plane wave incidence is a coherent sum of individual fields in response to the current components $Jm$⁠, which can be computed in a highly parallel fashion.

We combine the axisymmetric Maxwell solver with topology optimization to design a fully 3D single-piece plan-achromat (Fig. 4), rigorously taking into account off-axis angular aberrations. The lens has a diameter of $50λ$ and a thickness of $12λ$ with an NA of 0.12. It is optimized for aberration-free focusing at five frequencies and five angles (N = 25) within 23% bandwidth and 16° field of view. The lens achieves diffraction-limited focusing with SR >0.86 for all the design frequencies and angles, while the average absolute efficiency is 72%. To accurately model the off-axis angles, we consider up to 50 angular mode numbers so that the truncation error is less than 1% [compare Eqs. (5) and (6)], distributing our simulations over $∼1000$ CPUs with a computational time of about a week. Much larger diameters and numerical apertures can be designed by combining our approach with an overlapping domain method.²⁷ 

While we have shown that it is possible to dramatically reduce a cm-scale multiplet plan-achromat to a micrometer-scale single-piece with nanoscale 3D structuring, the overall system may still remain undesirably bulky because of the free space between the lens and the sensor.⁴⁰ We note that to further reduce this distance is not a simple matter of just designing a lens with a shorter focal length because conventional ray optics dictates that in order to achieve a desired magnification at a finite sensor resolution, the lens and sensor must be sufficiently separated. Conventionally, “compact imitations” such as a telephoto array are used to mimic the effect of a long-focus lens within a physically shorter device. On the other hand, it should be clear that the actual distance between the optical component and the sensor should not matter as long as the sensor measures the same output for any given input in the compact imitation as in the long-focus original. Since an arbitrary input can be decomposed into a sum of plane waves, the compact imitation must reproduce, for each input angle θ, a corresponding focal spot at the same position on a near sensor as the original would produce on a far sensor, i.e., the focal spot for θ must be formed at $f tan θ$ away from the sensor center where f is the virtual focal length of a desired long-focus lens, which may be longer than the actual lens-sensor distance $f′>f$ in the compact equivalent [Fig. 5 (top left inset) and also Fig. 1(c)]. This effectively means that any nanostructured equivalent, which labors to shorten the lens-sensor distance, must stretch the output angle as specified by the relation $θ′=tan−1(ff′tan θ)$⁠. (Optical etendue is conserved because the stretched angular output is compensated by a reduced numerical aperture that is smaller than what is implied by the physical width and lens-to-sensor distance.) Our inverse design formulation is naturally suited to design for such effects because we can explicitly specify an arbitrary input–output relation to our device.⁴¹ Here, we note that recently, an alternative approach to squeezing free space has also been proposed based on non-local metamaterials;^40,42 in contrast, our approach specifically considers the space-squeezing problem in an integral fashion, simultaneously with the focusing problem, and seeks to discover an “all-in-one unibody” nanophotonic space-squeezing plan-achromat.

Figure 5 shows a proof-of-concept 2D plan-achromat space-squeezer with a virtual focal length of $200λ$ but a physical lens-to-sensor distance of $130λ$ (54% compression). The lens is $50λ$ wide and $12λ$ thick, with a virtual NA of 0.12, and is optimized to perform at 10 frequencies and 10 angles over spectral and angular bandwidths of 23% and 16°, respectively, achieving virtual SR >0.8 for all the design frequencies and angles (note that the virtual SR is defined relative to the virtual airy disk, i.e., the ideal airy disk that would have been formed at the virtual focal distance of $200λ$⁠). One trade-off we found in the space-squeezer design is that the bigger the compression, the smaller the absolute focusing efficiency even for a small numerical aperture. While additional optimization constraints may be used to improve the transmission efficiency, we speculate that there exists a fundamental trade-off between compression and transmission—a theoretical question we will seek to elucidate in future work. On the other hand, such a trade-off may be entirely circumvented in an end-to-end framework, in which a front-end photonic design is fully coupled with a back-end computational reconstruction algorithm.⁴³ 

In summary, we have presented computational designs for single-piece plan achromats with SR >0.8 and focusing efficiencies up to 65%. We have also demonstrated that by solving the Maxwell equations under axial symmetry, one can design 3D axisymmetric plan-achromats that fully consider off-axis propagation. Our solutions also suggest the possibility of further reducing the lens-to-sensor distance by manipulating the relation between input and output angles and mimicking a longer virtual focal length.

Our optimizations were performed over $∼1000$ CPUs within one day to a week. We surmise that it is possible to design high-NA single-piece plan-achromats with millimeters in diameter and $∼100 μm$ in thickness, given sufficient computational resources on the order of $∼3000$ CPUs combined with advanced numerical solvers, which exploits the overlapping domains technique.²⁷ 

Our theoretical results demonstrate the untapped potential of volumetric nanophotonics and set the foundations for future works, which will be devoted to developing 3D devices tailored for specific fabrication technologies. For any given fabrication technology, there are a wealth of computational techniques to impose corresponding design constraints on feature sizes,²⁰ connectivity,¹⁹ and mechanical stability.¹⁸ For our proof-of-concept designs in this paper, it was convenient to parameterize the structures in terms of variable-height layers, but this is merely an artifact of this parameterization and is not fundamental to plan-achromat functionality. We expect that any degradation in performance arising from imposing fabrication constraints may be offset by increasing the device volumes (optimizing over even larger design spaces). Furthermore, broadband devices tend to be inherently robust to small manufacturing defects,⁴⁴ but standard techniques are available to incorporate additional robustness into TO if needed.¹⁰ 

Experimentally, there has been a surge of interest in fabricating multilayer metaoptics devices.^45,46 These designs require fabrication techniques more advanced than traditional CMOS single-step lithography, such as multistep lithography or laser writing techniques. Recent works suggest that 3D nanofabrication is poised to enable enhanced capabilities in 3D nanophotonics such as 3D interconnects,¹⁶ fiber couplers,¹⁷ and metasurfaces.^14,15 A particularly promising platform for volumetric metaoptics is the recently demonstrated implosion fabrication technique,¹³ which has the potential for realizing relatively constraint-free 3D geometries (such as multimaterial high-index structures held in place by a low-index background matrix) with feature sizes as small as a few tens of nanometers. Our work motivates further efforts to take advantage of emerging advances in 3D fabrication by demonstrating the potential of inverse-designed 3D metaoptics.

We note that our results represent an all-optical solution to imaging that does not require extensive calibration measurements. An alternative strategy that can ease the burden on optics is to measure “imperfect” point-spread functions and apply image processing algorithms to extract a faithful image.⁴⁷ Recently, we have also shown that nanophotonic structures can be inverse-designed in a holistic end-to-end fashion together with any image processing backend.⁴³ Ultimately, we expect that an optimal mix of hardware and software can be found to solve the most challenging problems in modern optical technologies and to pave the way for extraordinary functionalities that have not yet been explored.

We thank John Joannopoulos for helpful discussion. This work was supported in part by Villum Fonden through the NATEC (NAnophotonics for TErabit Communications) Centre (Grant No. 8692), the Danish National Research Foundation through NanoPhoton Center for Nanophotonics (Grant No. DNRF147), the U.S. Army Research Office through the Institute for Soldier Nanotechnologies (Award No. W911NF-18-2-0048), and the MIT-IBM Watson AI Laboratory (Challenge No. 2415).