Abstract
Metaoptics has achieved major breakthroughs in the past decade; however, conventional forward design faces challenges as functionality complexity and device size scale up. Inverse design aims at optimizing metaoptics design but has been currently limited by expensive bruteforce numerical solvers to small devices, which are also difficult to realize experimentally. Here, we present a general inversedesign framework for aperiodic largescale (20k × 20k λ^{2}) complex metaoptics in three dimensions, which alleviates computational cost for both simulation and optimization via a fast approximate solver and an adjoint method, respectively. Our framework naturally accounts for fabrication constraints via a surrogate model. In experiments, we demonstrate aberrationcorrected metalenses working in the visible with high numerical aperture, polychromatic focusing, and large diameter up to the centimeter scale. Such largescale metaoptics opens a new paradigm for applications, and we demonstrate its potential for future virtualreality platforms by using a metaeyepiece and a laser backilluminated microLiquid Crystal Display.
Similar content being viewed by others
Introduction
Metaoptics, a new class of planar optics, has reshaped the engineering of electromagnetic waves by using artificial subwavelength components or “metaatoms”^{1,2,3,4,5,6}. Recent breakthroughs in the physics^{7,8,9,10,11} and advancements in largescale metaoptics fabrication^{12,13,14} inspire a vision for a future in which metaoptics will be widely used. Recent studies have demonstrated cuttingedge technologies based on metaoptics platforms, such as polarization/lightfield/depth imaging cameras^{15,16,17,18}, metasurfacedriven OLEDs^{19}, virtual/augmented reality systems^{20,21}, compact spectrometers^{22,23,24}, etc. So far, the mainstream design of the metaoptics is mostly based on a “forward” methodology, in which one engineers each individual metaatom component (as a phase shifter) independently, according to a predefined phase profile^{25,26}. Forward design has demonstrated success in realizing simple device functions, like singlewavelength wave bending^{27,28,29} or focusing;^{30,31} however, it heavily relies on a priori intuitive knowledge and limits the development of largescale complex metaoptics that can realize multiple custom functions depending on wavelengths, polarizations, spins, and angles of incident light. As the complexity, diameter, or constraints of a design problem scale up, the ability of a forwarddriven method to search for an optimal solution becomes weaker and weaker. The future advancement of metaoptics demands a breakthrough in design philosophy.
In contrast to forward design, inverse design starts with desired functions and optimizes design geometries using computational algorithms. It has been a useful tool in solving largescale complex engineering problems such as optimizing the shape of bridges or aircraft wings. In recent years, inverse design has been reshaping the landscape of photonics engineering. Multiple flavors of inverse design techniques have been studied: topological optimization techniques, which use a local gradientbased optimization tool to search for optimal photonic geometries^{32,33}; and, machinelearning techniques^{34,35,36}, which train a neural network to find a design for a given response^{37} or train a generative network (e.g., generative adversarial network) to sample the highperformance designs^{38}. A recent evolution of inverse design in photonics optimizes the geometry and the postprocessing parameters endtoend^{39,40,41}. Inverse design has demonstrated significant success in optimizing photonic crystals^{42}, onchip nanophotonics^{43,44}, metasurfaces^{45,46}, and other devices.
Inverse design remains very challenging for aperiodic largescale metaoptics. The optimization relies on many iterations of simulations, which become computationally intractable as design dimension scales up due to the multiscale nature of design problems^{47}: the nanoscale metaatom (nm) and the macroscale metaoptics (100 s of µm to cm). On the one hand, it is unrealistic to model an aperiodic 3D device with a 1cm diameter using the finitedifference timedomain (FDTD) or the finite element analysis method, which can capture physics at nanoscale but are limited by both computation time and memory capacity. For example, it takes ~100 h in time and ~100 gigabytes in RAM memory for a FDTD solver to simulate a metasurface device of 50 µm^{2} in size (assuming a 5nm mesh size). On the other hand, raytracing simulations, which are suitable for largescale optics design, cannot capture the full wave nature of the optical field. They also only allow slowly varying phase profiles, excluding the rich physics of rapidly varying phase wavefronts offered by engineered metaatoms. To our knowledge, the diameter of inversedesigned fully threedimensional metasurfaces has been limited to about 200λ^{48,49,50,51}, about 100 µm for visible light. In addition, our inversedesign framework handles fabrication constraints inside a surrogate model, in contrast with most inversedesign frameworks, which need to add these constraints during optimization^{52}.
In this paper, we present a generic inversedesign framework that enables aperiodic largescale threedimensional complexfunction metaoptics compatible with fabrication constraints. Our inversedesign method is computationally tractable (requiring only a few hours using a desktop singlecore CPU) and advantageous for macroscale (>1000 s of λs) metaoptics design in tandem with exploitation of physics at the nanoscale. It greatly expands optical design to an unprecedented regime where conventional forward design is of limited use. The present design framework handles threedimensional simulations with six orders of magnitude more parameters than the proofofconcept twodimensional work^{53}. It controls the full polarization in contrast with ref. ^{21}, which is fundamentally limited to polarizationconverted light from lefthanded circularly polarized (LCP) state to righthanded circularly polarized (RCP) state. These unique inverse design features enable experimental demonstration of metaoptics with high numerical aperture (NA = 0.7) and complex functionality. For example, we show polarizationinsensitive RGBachromatic metalenses and even polychromatic metalenses. These inversedesigned metaoptics realizes mm to cm scale aperture size, which corresponds to an increase of four orders of magnitude in area compared with the state of the art. To prove the potential of largescale metaoptics in applications, we further demonstrate a metaopticsbased virtualreality (VR) platform.
Results
Inverse design theoretical framework
Fundamentally different from conventional forward design, the philosophy of inverse design is to start with the goal and then optimize it given the application’s constraints. For the design of lenses, the goal is to maximize the intensity at the focal spot; that is, we maximize \(\,I({\vec{{{{{{\bf{x}}}}}}}}_{{{{{{\bf{target}}}}}}}{{{{{\boldsymbol{,}}}}}}\,\vec{{{{{{\bf{p}}}}}}})\) over a vector \(\vec{{{{{{\bf{p}}}}}}}\) of geometric parameters defining the metasurface, where \({\vec{{{{{{\bf{x}}}}}}}}_{{{{{{\bf{target}}}}}}}\) is the location of the focal spot^{53}. For polychromatic lens design, the objective function finds a satisfying locally optimal geometry that maximizes the minimum intensity across the design wavelengths: \({\max }(\mathop{{{\min }}}\nolimits_{\lambda \in {\lambda }_{s}}({I}_{\lambda }({\vec{{{{{{\bf{x}}}}}}}}_{{{{{{\bf{target}}}}}}},\vec{{{{{{\bf{p}}}}}}})))\), where λ_{s} is a discrete set of wavelengths of interest and I_{λ} is the intensity function for a wavelength λ^{53}. This maximizes the focal intensity at multiple wavelengths simultaneously. We further reformulate this function to be differentiable as shown in the SI.
Fast and accurate “forward” evaluation of metaoptics performance is key to largescale inverse design. We introduce a threedimensional (3D) fast approximate solver that is based on the convolution of local fields and Green’s function (Fig. 1a). Accurate local fields above a training set of metaatoms are computed in advance using rigorous coupled wave analysis (RCWA). A surrogate model, which is based on Chebyshev interpolation^{54}, is then built to rapidly predict the local field of an arbitrary metaatom with fabricable parameters (SI). Our surrogate model is six orders of magnitude faster than a direct simulation using RCWA (SI). It also uses Chebyshev regression (leastsquare smoothing) to avoid artificial oscillations (SI)^{54}. By the equivalence principle, we convert the local fields to “artificial” sources of magnetic current density \({\vec{{{{{{\bf{S}}}}}}}}_{{{{{{\bf{local}}}}}}}\left(\vec{{{{{{\bf{x}}}}}}}{{{{{\boldsymbol{,}}}}}}\,\vec{{{{{{\bf{p}}}}}}}\right)\), and the focal intensity is computed by using a convolution between the current sources and vectorial Green’s function (Eq. (1))^{53}:
where \(\vec{{{{{{\bf{E}}}}}}}\big({\vec{{{{{{\bf{x}}}}}}}}_{{{{{{\bf{target}}}}}}}\big)\) is the electric field at the focal spot, ⊙ represents the Hadamard product, and \(\mathop{{{{{{\bf{G}}}}}}}\limits^{\leftrightarrow}\big(\vec{{{{{{\bf{x}}}}}}}{{{{{\boldsymbol{,}}}}}}\,{\vec{{{{{{\bf{x}}}}}}}}_{{{{{{\bf{target}}}}}}}\big)\) is the dyadic Green’s function from a local position \(\vec{{{{{{\bf{x}}}}}}}\) to a target position \({\vec{{{{{{\bf{x}}}}}}}}_{{{{{{\bf{target}}}}}}}\). Note that the Green’s function only needs to be computed once and can be reused in subsequent optimization iterations. It is analytical in free space and does not require a paraxial approximation. Here, we use a local periodic approximation (LPA) to predict the local fields, assuming neighboring metaatoms are similar^{53,55}. LPA is validated for our design by the agreement of predicted and experimental results; this is expected in the moderateNA regime where the metaatoms vary slowly over most of the surface (further discussed in the SI). To further speed up our simulator, we impose cylindrical symmetry on the design parameters while retaining the tiling of the metaatoms in Cartesian coordinates (see SI). Unlike fully axissymmetric designs^{51}, however, on a subwavelength scale our metaatoms break cylindrical symmetry.
Optimization in a highdimensional design space, when \(\vec{{{{{{\bf{p}}}}}}}\) is of dimension » 1000, is another challenge for inverse design. Here, we use a local gradientbased optimization method, called a “conservative convex separable approximation”^{56}, to search for an optimal design consisting of 10^{6} to 10^{9} degrees of freedom. We also applied a multistart approach by exploring multiple random initial design parameters^{57}. In our case, an initial design using the phasematching method is not possible since the aperture of the metalens is so large (up to 20k λ in diameter) that the required group delay^{11} to simultaneously satisfy the phase profiles of multiple wavelengths is three orders of magnitude larger than what a singlelayer metaatom can provide. For fast computation of the gradients \({\nabla }_{{{{{{\bf{p}}}}}}}I\big({\vec{{{{{{\bf{x}}}}}}}}_{{{{{{\bf{target}}}}}}}\big)\), we take advantage of an adjoint method^{58}, which can evaluate the gradients for all parameters simultaneously using only two simulations (Eq. (2)). In comparison, a traditional bruteforce method needs (N + 1) simulations, where N is the dimension of \(\vec{{{{{{\bf{p}}}}}}}\). The adjoint method is illustrated in Fig. 1b (details in the SI):
where ℜ denotes the real part, \(\mathop{{{{{{\bf{G}}}}}}}\limits^{\leftrightarrow}\big(\vec{{{{{{\bf{x}}}}}}}{{{{{\boldsymbol{,}}}}}}\,{\vec{{{{{{\bf{x}}}}}}}}_{{{{{{\bf{target}}}}}}}\big)\) is the dyadic Green’s function from a target position \({\vec{{{{{{\bf{x}}}}}}}}_{{{{{{\bf{target}}}}}}}\) to a local position \(\vec{{{{{{\bf{x}}}}}}}\), and \({{\nabla }_{{{{{{\bf{p}}}}}}}\vec{{{{{{\bf{S}}}}}}}}_{{{{{{\bf{local}}}}}}}\left(\vec{{{{{{\bf{x}}}}}}}{{{{{\boldsymbol{,}}}}}}\vec{{{{{{\bf{p}}}}}}}\right)\) is the gradient of the local current source with respect to the design parameter \(\vec{{{{{{\bf{p}}}}}}}\), which can also be fast evaluated by using a pretrained surrogate model at low cost. It means the gradient \({\nabla }_{{{{{{\bf{p}}}}}}}I\big({\vec{{{{{{\bf{x}}}}}}}}_{{{{{{\bf{target}}}}}}}\big)\) can be efficiently obtained everywhere at once in a backward simulation using an equivalent source \(\big({\vec{{{{{{\bf{E}}}}}}}{\big({\vec{{{{{{\bf{x}}}}}}}}_{{{{{{\bf{target}}}}}}}\big)}^{{{{{{\boldsymbol{* }}}}}}}{\nabla }_{{{{{{\bf{p}}}}}}}\vec{{{{{{\bf{S}}}}}}}}_{{{{{{\bf{local}}}}}}}\left(\vec{{{{{{\bf{x}}}}}}}{{{{{\boldsymbol{,}}}}}}\vec{{{{{{\bf{p}}}}}}}\right)\big)\). The gradient information was then fed into the optimizer for metadesign update (Fig. 1d). The whole design flow is summarized in Fig. 2. We started from a random metadesign and went through iterations of optimization loops, relying on a forward simulator and an adjoint simulator, until the device performance converged and met the design criteria. We then evaluated the final design in simulations and further in experiment.
Largescale inversedesigned polychromatic metalenses
Engineering a focus at multiple wavelengths and in different polarization states simultaneously is challenging, especially in the case of high NA. By applying the inversedesign method, we first demonstrated a polarizationinsensitive, RGBachromatic metalens. This metalens has a diameter of 2 mm and numerical aperture (NA) of 0.7. Figure 3a is an optical microscope image of the device fabricated using electron beam lithography (EBL) and atomic layer deposition^{59}. The inset scanning electron microscope image shows anisotropic TiO_{2} nanofin structures with spatially varying inversedesigned geometries on top of a fusedsilica substrate. The height of the nanofins is 600 nm and the squarelattice periodicity is 400 nm. Each nanofin has a rectangular shape whose sizes are determined by optimization and is aligned parallel to the unitcell axes. It partially converts LCP light to RCP light, and vice versa (Fig. 1c). The polarization conversion from L (σ_{−}〉) to R (σ_{+}〉) and R to L is equal by symmetry in our case, as described by the Jones’ matrix (Eq. (3)):
Where \({\widetilde{{{{{{\bf{t}}}}}}}}_{{{{{{\bf{L}}}}}}}\) and \({\widetilde{{{{{{\bf{t}}}}}}}}_{{{{{{\bf{s}}}}}}}\) are complex transmission along long and short axis, respectively, α is the rotation angle of nanofin, “out” means output field, and “in” means input field. Due to this symmetry and the fact that any polarization state can be written as superposition of LCP and RCP fields, our metalens design can focus light equally well for any arbitrary polarization state^{59,60}. Figure 3b shows the simulation results for the focal intensity distribution along the optical axis at the design RGB wavelengths of 488, 532, and 658 nm. These wavelengths are chosen to correspond to our singlewavelength laser diodes. The inset is the zoomedin view of the focal peaks, which shows achromatic focusing with negligible focal shifts (<50 nm). Figure 3c is the measured focal intensity distribution at the RGB wavelengths in the XZ plane, where X is along the lens radial direction and Z is along the optical axis. The maximum focal shift is ~1.5 µm, which is ~0.15% of the focal length. Figure 3d, from top to bottom, is the measured intensity distribution at the focal planes of the blue, green, and red wavelengths, respectively. Their respective measured focal intensity profiles (Fig. 3e) imply diffractionlimited focus (detailed analysis can be found in the SI). We measured the absolute focusing efficiency, which is defined as the ratio between the power in the focal spots and the incident power, as a function of the incidence polarization angles. Figure 3f shows that the absolute efficiency is about 15% at RGB wavelengths and is independent of the polarization angle of the incident light. Moreover, our metalens focuses light of an arbitrary polarization state to its orthogonal state, which is useful for improving the imaging contrast. We further characterized the imaging performance of the metalens using the United States Air Force (USAF) resolution target. Figure 3g–i is the imaging result of the element No. 5 and No. 6 from group No. 7 under blue, green, and red illumination. The smallest feature size is 2.2 µm and can be clearly resolved. To demonstrate achromatic imaging, we further imaged the same area using synthesized whitelight illumination by mixing RGB color in the incident light. The result is a clear whitish image with the same magnification (Fig. 3j). More imaging results under other synthesized light illumination can be found in SI.
The inversedesign method has more pronounced advantages over conventional forward design methods when designing a metaoptics with more complicated functions. Forward design methods, like wavefront phasematching, struggle in the regime where no single metaatom can simultaneously satisfy the targeted phase profiles for multiple functions. They cannot balance compromises between metaatoms systematically because they optimize each metaatom separately. A good phasematching will try to reduce the overall phase errors, at the risk of omitting one of the functions as well as ignoring crosstalks between functions. They also neglect the effect of a nonuniform amplitude or phase profile. A good phasematching sometimes comes at the cost of low efficiency due to the intrinsic correlation between the phase and amplitude of the engineered electromagnetic wave by metaatoms. Moreover, forward designs are usually oneway without feedback loops, and thus do not provide confirmation of optimality or robustness. Importantly, forward designs require a priori knowledge of the desired wave solution, which is unavailable for complex problems. In comparison, our inversedesign method can obtain previously unknown solutions to complex design problems because it starts only with the design objective and iteratively searches for an optimal solution in a hyperdimensional design space. It also evaluates the objective functions directly against design parameters and balances the nonuniform amplitude/phase profiles automatically across the metasurface to optimize complex objective functions and crosstalks.
To prove the concept, we further demonstrated an experimental polychromatic metalenses with sixwavelengthachromaticfocusing performance for visible light. These two metalenses have an aperture diameter of 2 mm and NAs of 0.3 and 0.7. Figure 4a is the SEM image of the NA = 0.3 metalens. This metalens is designed for achromatic focusing at six wavelengths of 490, 520, 540, 570, 610, and 650 nm. Figure 4b is the simulation result showing their focal intensity distribution along the optical axis. The measured focusing intensity (Fig. 4c) in the XZ plane shows good agreement with the simulation results. The maximum focal shift among design wavelengths is 500 nm (<0.02% of the focal length). The measured focusing efficiency is ~8%, and imaging results are shown in the SI. The simulation and measurement results of the NA = 0.7 metalens are also shown in the SI. Figure 4d shows that the measured fullwidthhalfmaximums of the focal spots in comparison with the ideal Airyfunction theory. The subtle differences are because we used a supercontinuum laser as the light source, which has a larger linewidth (FMHWs) of ~5 nm in comparison with ~0.5 nm linewidth of laser diodes (SI). Figure 4e–j is the measured focal intensity distribution at a common focal plane of six design wavelengths. The measurement results of the NA = 0.7 metalens are shown in the SI.
To further prove the scalability of our inversedesign method, we designed and fabricated a cmscale metalens. This metalens is designed for achromatic focusing at RGB wavelengths with an NA of 0.3. Figure 5a shows the 1cmdiameter RGBachromatic flat metaoptics on 2inch fused silica wafer with a reference ruler behind. The inset is the SEM image showing the metaatoms building blocks. We utilized a fast Ebeam writer and operated at a high current. Consequently, we achieved 10nm structural resolution at a low cost in fabrication time. Figure 5b is the simulation result showing the focal intensity distribution along the optical axis at design wavelengths, and the inset is the zoomedin view of the peaks to show its achromatic focusing performance. The measured focal intensity distribution in the XZ plane is shown in Fig. 5c. The maximum focal shift among RGB wavelengths is ~4.5 µm, which is ~0.03% of the design focal length. Figure 5d–f shows the measured focal intensity distribution at the focal planes of λ = 488 nm, 532 nm, and 658 nm. The measured focusing efficiency at RGB is around 15%, and the design simulations show ~24% (SI). The difference can be attributed to fabrication errors. For example, the stitching errors between writing fields result in reduced focusing efficiency. We discuss this result and strategies for further improvement in the final section. The slight distortion of the focal spots is due to the nonuniform incident illumination over the cmscale lens’ aperture, which was not anticipated during design. We further characterized the metalens by imaging the whole group No. 7 of the USAF resolution targets. Figure 5g–i is the imaging result under illumination of blue, green, and red incident light, respectively, which shows excellent imaging performance.
Furthermore, we compare forward designs with our inverse design using the 1cmdiameter RGBachromatic polarizationinsensitive metalens as a benchmark, and the simulation results are summarized in SI. The forward design results vary with the definitions of objective functions that quantify the phasematching conditions. The corresponding focusing efficiencies are not only lower but also nonuniform at RGB wavelengths. It reveals the limitations of forward design when applied to a design problem that involves multiple objectives and is subject to multiple constraints. In comparison, the inverse design results show better (~24%) and uniform focusing efficiencies (SI). Furthermore, the inverse design can be used to mitigate ghost focal spots or reduce halo. For example, its objective function can define the light intensity distribution along the optical axis or the scattering of zerothorder light.
Virtualreality imaging demonstration
Largescale metaoptics may have significant impact on many applications. Here, we demonstrate a VR imaging system based on our metaoptics. VR is a technology that creates an immersive experience by replacing reality with an imaginary world^{61}. Its recent breakthroughs have not only attracted attention from the scientific community and industry but have also piqued the interest of the general public. Unfortunately, widespread use of VR devices has been hindered by a bottleneck in the optical architecture. The eyepieces used in current VR headsets mostly rely on refractive singlets, which suffer from bulky size and weight, and they furthermore compromise the viewing experience due to spherical and chromatic aberrations^{62}. Metaoptics offer a technology to address these challenges of current VR systems^{21}.
Figure 6a is the schematic of our VR system, based on our cmscale RGBachromatic metaeyepiece and a laserilluminated microLCD. The microLCD is placed close to the focal plane of the metaeyepiece, and the image on the display is projected via the metaeyepiece onto the retina, creating a virtual scene. In the experiment, we used a tube lens to mimic the cornea and eye crystalline lens and a CMOS camera to mimic the retina. In addition, we homebuilt a neareye display using the laser light as the backillumination source. Such a display offers high brightness and a wide color gamut due to the narrow linewidth. The pixel size is about 8 µm, matching the state of the art. Figure 6b shows the key components of the metaeyepiece and the display as illustrated in the dashed brown box of Fig. 6a. We first demonstrate binary VR imaging. Figure 6c shows the VR image of a red letterH shield logo, and Fig. 6d is the zoomedin view of one corner (from the white dashed box of Fig. 6c). One can see that the metaeyepiece resolves every pixel of the display. Figure 6e, f is the imaging result for an MIT logo under green and blue illumination, respectively. We further demonstrated grayscale VR imaging. Figure 6g, h is a grayscale imaging result (in red light) showing a Harvard building and statue, respectively. Figure 6i, j shows the grayscale VR images of a building and lighthouse in green and blue, respectively. These RGBcolor imaging results imply an ability to image in fullcolor, because color images are simply formed by mixing these primary colors. For example, Fig. 7a–c shows VR imaging of distinct red, green, and blue circles, respectively. Figure 7d is the simulated color VR imaging result by superimposing Fig. 7a–c, which show synthesized colors of yellow, magenta, cyan, and white in the circle overlapping regions. Furthermore, Fig. 7e–g shows VR imaging of a Harvard tower in red, green, and blue channel. Figure 7h is the simulated fullcolor imaging result by superimposing the RGB images (Fig. 7e–g). In addition to the static VR images, our VR system can also display a dynamic VR object. Figure 7i–l displays a running cat that is captured at 0, 180, 460, and 600 ms, respectively. The neareye display has a refresh rate of 60 Hz, and the recorded movie can be found in the SI. We further discuss a strategy to reduce form factor of the display by using a metasurfacebased illumination plate (SI).
This work shows major advances over the previous VR system^{21}. Thanks to the innovative inversedesign method, the metaoptics has increased the aperture size from mm to cm, which means it can be integrated with micro displays and is more realistic for applications. Micro displays are the future trend for VR optical engines; however, there has not yet been an eyepiece solution that can resolve highresolution (~5 µm) color images. Second, the metaoptics now has polarizationinsensitive focusing performance, which alleviates additional polarizationselection components (e.g., linear polarizer and phase retarder) and makes better use of incident light (focusing efficiency increases by more than double compared to ref. ^{21}). Third, the metaatoms now have a simple geometry shape and, thus, are more compatible with largescale and mass production. Finally, displaying a movie is now possible thanks to the high refresh rate of our system. In the future, we believe metaoptics will augment conventional lens platforms^{63} to form a highperformance aberrationfree compact hybrid eyepiece for VR/AR. The metaeyepiece in this work corrects both chromatic and monochromatic aberrations under normal incidence. Future work will improve upon this to optimize corrections for higherorder aberrations such as coma and field of curvature. Possible research directions include a metasystem consisting of multiple metasurfaces pieces^{47} or a hybrid design^{63} that combines a refractive element with metaoptics.
Discussion
In this paper, we presented a general inversedesign framework that is suitable for the largescale 3D photonicdevice optimization. We demonstrated inversedesigned 3D metaoptics of large diameters, including 2mmdiameter RGBachromatic and polychromatic metalenses, and even a cmscale RGBachromatic metalens, the largest to date, which consists of ~10^{9} metaatoms. Furthermore, we demonstrate a path toward a future VR platform based on a metaeyepiece and a laserilluminated microLCD. This inversedesign method is also applicable to optimizing other optical elements in a VR/AR system, such as optical combiners.
In general, the development of nextgeneration wearable imaging platforms that have small form factor, high focusing efficiency, and correct multiple aberrations remains a challenging research topic. A recent developed “pancake lenses” for VR headsets^{64} comprising a concave halfmirror and a reflection polarizer is more compact compared to a conventional refractive eyepiece; however, the transmission efficiency is limited to ~12.5%. Our demonstrated metaoptics so far has a focusing efficiency of ~15% at RGB wavelengths under unpolarized illumination. In comparison, our previously reported 2mmdiameter RGBachromatic polarizationsensitive metalens (NA = 0.7) showed ~12% focusing efficiency at RGB wavelengths under LCP illumination, which is equivalent to ~6% under unpolarized illumination. Polarizationinsensitive focusing of our metalens is achieved by using anisotropic metaatoms. It means that the imaging contrast can be improved by selecting output light polarization despite relatively low focusing efficiency. In comparison, isotropic metaoptics is not ideal for multiwavelength engineering and direct imaging applications since it suffers from background light when focusing efficiency is low^{13}. To reduce the power consumption of a future VR device, the focusing efficiency of our metaeyepieces needs to be increased. Further device performance improvements require innovations in the metaatoms, i.e., in the building blocks of the metasurface. We envision the nextgeneration of freeform and multiplelayered metaatoms, which embraces more degrees of freedom and richer physics, as the key to greater performance and functionality. Implementing complex metaatoms in a largearea inversedesign framework also requires advances in computational methods. The Chebyshev surrogate model used in this work needs an exponentially increasing dataset for more design parameters, but recent work has shown that neural networks utilizing new activelearning techniques^{65} and incorporating physics knowledge^{66} can handle ten parameters with orders of magnitude less training data. These advances mean that future surrogatebased fast solvers can use more accurate methods based on supercell domains^{66} that better capture rapid surface variations. Fully freeform topology optimization has also begun to steadily approach larger scales by exploiting domaindecomposition approximations^{48,67} and axisymmetric restrictions^{51} with the help of largescale computing power and fast surrogate simulations^{68}. Besides engineering the light focusing of metalenses, one can also take advantage of inverse design to better exploit other physical processes, such as optical^{69} nonlinear effects, and to gain better understanding of the multiphysics phenomena in photonic platforms. We believe that a more and more important role will be played by largescale inversedesign methods in the future development of metaoptics.
Methods
Simulation
The metaatoms are simulated using the method of RCWA. In the simulation setup, the height of the TiO_{2} metaatoms is 600 nm, the periodicity of the unit cell is 400 nm, and the substrate is fused silica. The incident light is configured to LCP (RCP), and monitored light is in the opposite polarization state of RCP (LCP). The simulation wavelength sweeps from 480 nm to 680 nm in the visible.
Fabrication
The metalenses are fabricated on glass wafers. The fabrication starts with spincoating of resists in the following manner: a thinlayer of Hexamethyldisilazane (HMDS), a layer of 600nmthick electron beam resist (Zeon Specialty Materials, ZEP520A), and then a final layer of conductive polymer (Showa Denko, ESPACER 300) to dissipate charges during the following EBL process. After that, the 2mm diameter metalens samples are exposed using Elionix ELSF125 and the 1cm diameter metalens is exposed using Elionix HS50, respectively, followed by removal of the conductive polymer layer in water and development of exposed resist in oXylene solution, respectively. Next, a thin film of TiO_{2} is deposited onto the developed sample using lowtemperature atomic layer deposition (Cambridge Nanotech, Savannah). TiO_{2} thin film is conformally deposited on the sample not only completely filling inside the developed area but also on top of the remaining resist film. The excessively grown TiO_{2} layer is later removed using reactive ion etching (Oxford Instruments, PlasmaPro 100 Cobra 300) with etchant gases of CHF_{3}, O_{2}, and Ar, until the underlying resist layer is exposed. In the final step, the resist layer is stripped off in solution of Remover PG (Kayaku Advanced Materials) at 85 °C for 24 hours, leaving only the TiO_{2} nanostructures on the glass wafer.
Data availability
The metaatom libraries that are used to design the metasurfaces in this study have been deposited in the Harvard University internal database. The data are available under restricted access for noncommercial use, and access can be obtained from the corresponding authors upon reasonable request.
Code availability
The codes that support the findings of this study are available upon reasonable request from the corresponding authors.
Change history
15 September 2023
In this article the hyperlink provided for reference 69 was incorrect. The original article has been corrected.
References
Yu, N. & Capasso, F. Flat optics with designer metasurfaces. Nat. Mater. 13, 139–150 (2014).
Chen, H.T., Taylor, A. J. & Yu, N. A review of metasurfaces: physics and applications. Rep. Prog. Phys. 79, 076401 (2016).
Khorasaninejad, M. & Capasso, F. Metalenses: versatile multifunctional photonic components. Science 358, aam8100 (2017).
Lalanne, P. & Chavel, P. Metalenses at visible wavelengths: past, present, perspectives. Laser Photonics Rev. 11, 1600295 (2017).
Tseng, M. L. et al. Metalenses: advances and applications. Adv. Optical Mater. 6, 1800554 (2018).
He, Q., Sun, S., Xiao, S. & Zhou, L. Highefficiency metasurfaces: principles, realizations, and applications. Adv. Optical Mater. 6, 1800415 (2018).
Wang, S. et al. Broadband achromatic optical metasurface devices. Nat. Commun. 8, 1–9 (2017).
Shrestha, S., Overvig, A. C., Lu, M., Stein, A. & Yu, N. Broadband achromatic dielectric metalenses. Light. Sci. Appl. 7, 1–11 (2018).
Wang, S. et al. A broadband achromatic metalens in the visible. Nat. Nanotechnol. 13, 227–232 (2018).
Chen, W. T. et al. A broadband achromatic metalens for focusing and imaging in the visible. Nat. Nanotechnol. 13, 220–226 (2018).
Chen, W. T., Zhu, A. Y. & Capasso, F. Flat optics with dispersionengineered metasurfaces. Nat. Rev. Mater. 5, 604–620 (2020).
She, A., Zhang, S., Shian, S., Clarke, D. R. & Capasso, F. Large area metalenses: design, characterization, and mass manufacturing. Opt. Express 26, 1573–1585 (2018).
Park, J.S. et al. Allglass, large metalens at visible wavelength using deepultraviolet projection lithography. Nano Lett. 19, 8673–8682 (2019).
Hu, T. et al. CMOScompatible aSi metalenses on a 12inch glass wafer for fingerprint imaging. Nanophotonics 9, 823–830 (2020).
CamaydMuñoz, P., Ballew, C., Roberts, G. & Faraon, A. Multifunctional volumetric metaoptics for color and polarization image sensors. Optica 7, 280–283 (2020).
Rubin, N. A. et al. Matrix Fourier optics enables a compact fullStokes polarization camera. Science 365, eaax1839 (2019).
Lin, R. J. et al. Achromatic metalens array for fullcolour lightfield imaging. Nat. Nanotechnol. 14, 227–231 (2019).
Guo, Q. et al. Compact singleshot metalens depth sensors inspired by eyes of jumping spiders. Proc. Natl Acad. Sci. USA 116, 22959–22965 (2019).
Joo, W.J. et al. Metasurfacedriven OLED displays beyond 10,000 pixels per inch. Science 370, 459–463 (2020).
Lee, G.Y. et al. Metasurface eyepiece for augmented reality. Nat. Commun. 9, 1–10 (2018).
Li, Z. et al. Metaoptics achieves RGBachromatic focusing for virtual reality. Sci. Adv. 7, eabe4458 (2021).
FarajiDana, M. et al. Compact folded metasurface spectrometer. Nat. Commun. 9, 1–8 (2018).
Zhu, A. Y. et al. Compact aberrationcorrected spectrometers in the visible using dispersion‐tailored metasurfaces. Adv. Optical Mater. 7, 1801144 (2019).
McClung, A., Samudrala, S., Torfeh, M., Mansouree, M. & Arbabi, A. Snapshot spectral imaging with parallel metasystems. Sci. Adv. 6, eabc7646 (2020).
Lalanne, P. Waveguiding in blazedbinary diffractive elements. JOSA A 16, 2517–2520 (1999).
Khorasaninejad, M. & Capasso, F. Broadband multifunctional efficient metagratings based on dielectric waveguide phase shifters. Nano Lett. 15, 6709–6715 (2015).
Zhang, L. et al. Ultrathin highefficiency midinfrared transmissive Huygens metaoptics. Nat. Commun. 9, 1–9 (2018).
Yu, N. et al. Light propagation with phase discontinuities: generalized laws of reflection and refraction. Science 334, 333–337 (2011).
Ni, X., Emani, N. K., Kildishev, A. V., Boltasseva, A. & Shalaev, V. M. Broadband light bending with plasmonic nanoantennas. Science 335, 427–427 (2012).
Aieta, F. et al. Aberrationfree ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces. Nano Lett. 12, 4932–4936 (2012).
Khorasaninejad, M. et al. Metalenses at visible wavelengths: diffractionlimited focusing and subwavelength resolution imaging. Science 352, 1190–1194 (2016).
Molesky, S. et al. Inverse design in nanophotonics. Nat. Photonics 12, 659–670 (2018).
Sell, D., Yang, J., Doshay, S., Yang, R. & Fan, J. A. Largeangle, multifunctional metagratings based on freeform multimode geometries. Nano Lett. 17, 3752–3757 (2017).
Ma, W. et al. Deep learning for the design of photonic structures. Nat. Photonics 15, 1–14 (2020).
Trivedi, R., Su, L., Lu, J., Schubert, M. F. & Vuckovic, J. Datadriven acceleration of photonic simulations. Sci. Rep. 9, 1–7 (2019).
Lu, L. et al. Physicsinformed neural networks with hard constraints for inverse design. SIAM J. Sci. Comput. 43, B1105–B1132 (2021).
An, S. et al. A deep learning approach for objectivedriven alldielectric metasurface design. ACS Photonics 6, 3196–3207 (2019).
Jiang, J., Chen, M. & Fan, J. A. Deep neural networks for the evaluation and design of photonic devices. Nat. Rev. Mater. 6, 1–22 (2020).
Lin, Z. et al. Endtoend nanophotonic inverse design for imaging and polarimetry. Nanophotonics 10, 1177–1187 (2021).
Tseng, E. et al. Neural nanooptics for highquality thin lens imaging. Nat. Commun. 12, 1–7 (2021).
Sitzmann, V. et al. Endtoend optimization of optics and image processing for achromatic extended depth of field and superresolution imaging. ACM Trans. Graph. 37, 1–13 (2018).
Dory, C. et al. Inversedesigned diamond photonics. Nat. Commun. 10, 1–7 (2019).
Sapra, N. V. et al. Onchip integrated laserdriven particle accelerator. Science 367, 79–83 (2020).
Piggott, A. Y. et al. Inverse design and demonstration of a compact and broadband onchip wavelength demultiplexer. Nat. Photonics 9, 374–377 (2015).
Campbell, S. D. et al. Review of numerical optimization techniques for metadevice design. Optical Mater. Express 9, 1842–1863 (2019).
Chung, H. & Miller, O. D. HighNA achromatic metalenses by inverse design. Opt. Express 28, 6945–6965 (2020).
Backer, A. S. Computational inverse design for cascaded systems of metasurface optics. Opt. Express 27, 30308–30331 (2019).
Phan, T. et al. Highefficiency, largearea, topologyoptimized metasurfaces. Light. Sci. Appl. 8, 1–9 (2019).
Bayati, E. et al. Inverse designed metalenses with extended depth of focus. ACS Photonics 7, 873–878 (2020).
Mansouree, M. et al. Multifunctional 2.5 d metastructures enabled by adjoint optimization. Optica 7, 77–84 (2020).
Christiansen, R. E. et al. Fullwave Maxwell inverse design of axisymmetric, tunable, and multiscale multiwavelength metalenses. Opt. Express 28, 33854–33868 (2020).
Vercruysse, D., Sapra, N. V., Su, L., Trivedi, R. & Vučković, J. Analytical level set fabrication constraints for inverse design. Sci. Rep. 9, 1–7 (2019).
Pestourie, R. et al. Inverse design of largearea metasurfaces. Opt. Express 26, 33732–33747 (2018).
Boyd, J. P. Chebyshev and Fourier Spectral Methods (Courier Corporation, 2001).
PérezArancibia, C., Pestourie, R. & Johnson, S. G. Sideways adiabaticity: beyond ray optics for slowly varying metasurfaces. Opt. Express 26, 30202–30230 (2018).
Svanberg, K. A class of globally convergent optimization methods based on conservative convex separable approximations. SIAM J. Optim. 12, 555–573 (2002).
Kan, A. R. & Timmer, G. T. Stochastic global optimization methods part I: clustering methods. Math. Program. 39, 27–56 (1987).
Strang, G. Computational Science and Engineering. (WellesleyCambridge Press, 2007).
Devlin, R. C., Khorasaninejad, M., Chen, W. T., Oh, J. & Capasso, F. Broadband highefficiency dielectric metasurfaces for the visible spectrum. Proc. Natl Acad. Sci. USA 113, 10473–10478 (2016).
Chen, W. T., Zhu, A. Y., Sisler, J., Bharwani, Z. & Capasso, F. A broadband achromatic polarizationinsensitive metalens consisting of anisotropic nanostructures. Nat. Commun. 10, 1–7 (2019).
Sherman, W. R. & Craig, A. B. Understanding Virtual Reality: Interface, Application, and Design (Morgan Kaufmann, 2018).
Beams, R., Kim, A. S. & Badano, A. Transverse chromatic aberration in virtual reality headmounted displays. Opt. Express 27, 24877–24884 (2019).
Chen, W. T. et al. Broadband achromatic metasurfacerefractive optics. Nano Lett. 18, 7801–7808 (2018).
Sneider, R. & Horne, J. Kopin Corporation Announces AllPlastic Pancake® Optics with Excellent Performance https://www.businesswire.com/news/home/20210615005440/en/ (2021).
Pestourie, R., Mroueh, Y., Nguyen, T. V., Das, P. & Johnson, S. G. Active learning of deep surrogates for PDEs: Application to metasurface design. npj Comput. Mater. 6, 1–7 (2020).
Pestourie, R., Mroueh, Y., Rackauckas, C., Das, P. & Johnson, S. G. Physicsenhanced deep surrogates for PDEs. Preprint at https://doi.org/10.48550/arXiv.2111.05841 (2021).
Lin, Z. & Johnson, S. G. Overlapping domains for topology optimization of largearea metasurfaces. Opt. Express 27, 32445–32453 (2019).
Chen, M. et al. Physicsaugmented deep learning for highspeed electromagnetic simulation and optimization. Preprint at https://doi.org/10.48550/arXiv.2203.01248 (2021).
Fisher, S., Pestourie, R. & Johnson, S. G. Efficient perturbative framework for coupling of radiative and guided modes. Preprint at https://doi.org/10.48550/arXiv.2203.02602 (2022).
Acknowledgements
Z.L., J.P., and F.C. are supported by the Defense Advanced Research Projects Agency (grant# HR00111810001) and AFOSR (grant # FA95502110312). This work was performed in part at the Center for Nanoscale System (CNS), a member of the National Nanotechnology Coordinated Infrastructure (NNCI), which is supported by the National Science Foundation under NSF award no. 1541959. CNS is part of Harvard University. R.P. was supported by the U.S. Army Research Office through the Institute for Soldier Nanotechnologies (Award No. W911NF1820048) and the MITIBM Watson AI Laboratory (Challenge No. 2415). Y.W.H. is supported by the National Research Foundation, Prime Minister’s Office, Singapore under its Competitive Research Program (CRP Award No. NRFCRP15201503). The authors thank Meredith Dost for her suggestions in manuscript editing.
Author information
Authors and Affiliations
Contributions
Z.L. and R.P. conceived the original concept. R.P. developed the inversedesign framework with contribution from Z.L. Z.L. conducted the metalenses fabrication, measurement, and virtual reality imaging experiment. J.P. and Y.W.H. contributed to the device fabrication and SEM imaging. S.J. and F.C. supervised the project. Z.L. and R.P. prepared the manuscript with input from the authors.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Peer review
Peer review information
Nature Communications thanks the anonymous reviewer(s) for their contribution to the peer review of this work.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Li, Z., Pestourie, R., Park, JS. et al. Inverse design enables largescale highperformance metaoptics reshaping virtual reality. Nat Commun 13, 2409 (2022). https://doi.org/10.1038/s41467022299733
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41467022299733
This article is cited by

Broadband spin and angle comultiplexed waveguidebased metasurface for sixchannel crosstalkfree holographic projection
eLight (2024)

Integral imaging neareye 3D display using a nanoimprint metalens array
eLight (2024)

Advances in information processing and biological imaging using flat optics
Nature Reviews Electrical Engineering (2024)

CostEffective and Environmentally Friendly Mass Manufacturing of Optical Metasurfaces Towards Practical Applications and Commercialization
International Journal of Precision Engineering and ManufacturingGreen Technology (2024)

Recent advancements of metalenses for functional imaging
Nano Convergence (2023)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.