High-efficiency high-NA metalens designed by maximizing the efficiency limit

Theoretical bounds are commonly used to assess the limitations of photonic design. Here we introduce a more active way to use theoretical bounds, integrating them into part of the design process and identifying optimal system parameters that maximize the efficiency limit itself. As an example, we consider wide-field-of-view high-numerical-aperture metalenses, which can be used for high-resolution imaging in microscopy and endoscopy, but no existing design has achieved a high efficiency. By choosing aperture sizes to maximize an efficiency bound, setting the thickness according to a thickness bound, and then performing inverse design, we come up with high-numerical-aperture (NA = 0.9) metalens designs with record-high 98% transmission efficiency and 92% Strehl ratio across all incident angles within a 60-deg field of view, reaching the maximized bound. This maximizing-efficiency-limit approach applies to any multi-channel system and can help a wide range of optical devices reach their highest possible performance.


I. INTRODUCTION
With the rapid advance of nanophotonic design and fabrication, there has been increasing interests in exploring the ultimate limit of device performance.In recent years, theoretical bounds have been proposed for the operational bandwidth [1][2][3][4][5], efficiency (e.g., reflection, transmission, absorption) [6][7][8][9][10], coupling strengths [11,12], energy concentration [13,14], and the device thickness [15,16].Conventionally, these bounds are used to assess how close a design approaches fundamental limits [17].One can then set realistic targets and avoid futile attempts on trying to design things that are theoretically impossible.While understanding the limit is important, it would be desirable if theoretical bounds can also play a more active role in improving the design and in pushing the limit itself.
To identify and to reach the highest possible performance, here we propose to integrate theoretical bounds into part of the design process.In the first step, we consider a recently introduced bound on the channelaveraged transmission efficiency [10], which depends on the sizes of the input and output apertures [D in and D out in Fig. 1(a)].Given any output aperture D out of interest, we identify the optimal input aperture size D opt in that maximizes the efficiency bound.In the second step, we set the thickness h based on a recently introduced bound on the minimal device thickness [15].In the third step, we perform inverse design using the optimal D in and the minimal h.Applying this approach to a metalens with NA = 0.9 with a 60 • FOV, we achieve a transmission efficiency of 98% and Strehl ratio of 92% across the full FOV, reaching the maximized efficiency limit.Previous inverse designs attained significantly lower efficiencies because without the guidance from theoretical bounds, they invariably adopted suboptimal geometry parameters.Both bounds we employ [10,15] follow from the target functionality of any linear multi-channel optical system, so the proposed strategy here is general, not limited to metalens designs.

II. NUMERICAL OPTIMIZATION METHOD
We first describe our inverse design methodology.We employ free-form topology optimization [49][50][51][52], allowing non-intuitive permittivity profiles.We have made our code open-source, available at Ref. [53].We consider the transverse magnetic waves (E x , H y , H z )(y, z) of a system invariant in the x dimension.Given the a-th incident angle θ a in within the FOV, the desired focal spot position is r a f = (y = f tan θ a in , z = f ) where f is the focal length.We quantify the focusing quality by the ratio between the intensity of the actual design at r a f and the focal spot intensity for an ideal lens (i.e., perfectly transmitting and FOV metalens designs and in this work (purple for monochromatic, green for achromatic).Details are listed in Supplementary Tab.S1.Solid lines show the theoretical limit [10] with Din = Dout and with the optimal Din = D opt in used here.
free of aberrations), as which is the product of the Strehl ratio (SR) and the transmission efficiency T at incident angle θ a in .The SR [48,54] is defined as the actual intensity at the focal spot divided by the peak intensity of an ideal diffractionlimited focus normalized by the total transmitted power of the metalens.The transmission efficiency T is the total transmitted power divided by the incident power.
We map out the focal intensity as a function of the incident angle through the generalized transmission matrix t = CA −1 B of the metalens [33].Here, matrix A = −∇ 2 −(2π/λ) 2 ε r (r) is the discretized Maxwell differential operator on E x (y, z) at wavelength λ for the metalens structure defined by its relative permittivity profile ε r (r).The a-th column of the M -column input matrix contains the source profile that generates an incident plane wave at the a-th incident angle θ a in within the input aperture.The b-th row of the M -row output projection matrix C = [C 1 ; • • • ; C M ] performs angular spectrum propagation [55] that propagates E x on the output surface of the metalens to position r b f on the focal plane.Then, the diagonal elements t aa are the field amplitudes E x (r a f ; θ a in ) at the focal spots in Eq. ( 1), for all incident angles {θ a in } within the FOV of interest.More details are given in Supplementary Secs.1-2.
To reach optimal I a = SR a • T a for all incident angles over the FOV, we maximize the worst-case min a I a within the FOV.To make the objective function differentiable, we cast the problem into an equivalent epigraph form [56] using a dummy variable g, max g,εr g, subject to g ≤ I a (ε r ). ( We perform the optimization using the gradient-based method of moving asymptotes [57] implemented in the open-source package NLopt [58].Under finite-difference discretization with grid size ∆x, the gradient of I a with respect to the permittivity profile ε r of the metalens is calculated via the adjoint method (see Supplementary Sec. 3) as where k 0 = 2π/λ is the free-space wave vector; operators •, * , and T stand for the element-wise Hadamard product, complex conjugation, and vector transpose respectively.We perform the computation with the open-source multichannel Maxwell solver MESTI [59] using single-precision arithmetic.
During the optimizations, we update the permittivity profile with a macropixel size of 4∆x = λ/10 to reduce the dimension of the design space and to keep the minimal feature size large while performing the simulations with a finer resolution of ∆x = λ/40 for accuracy.Since the desired response is symmetric, we impose ε r (r) to be mirror-symmetric with respect to the lens center y = 0 and only parameterize ε r of the macropixels in the left half of the metalens [red box in Fig. 3(a)] as the optimization variables [60].

III. VALIDATION OF THEORETICAL BOUNDS
Before integrating theoretical bounds into the design process, it is prudent to first verify whether the bounds we use are valid and if they are tight.Such a validation is nontrivial and requires inverse design, since intuitionbased designs are not likely to reach these bounds.For this validation, we consider a relatively small system: metalenses with NA = 0.9, FOV = 60 • , and output diameter D out = 16λ.
In Ref. [10], we derived that the transmission efficiency averaged over all input channels within the FOV, ⟨T ⟩, cannot exceed N eff /N in where N in is the number of input channels, and the inverse participation ratio i quantifies the effective number of high-transmission channels using the singular values {σ i } of the transmission matrix.After writing down the ideal transmission matrix of diffraction-limited wide-FOV metalenses in air (Supplementary Sec. 2), we plot N eff /N in as the black solid lines in Fig. 2. The efficiency bound N eff /N in depends on the diameter D in of the input aperture and has a local maximum at D in = D opt in ≈ 8λ.To verify this dependence, we perform the topology optimization of Eq. ( 2) for different D in , with the refractive index of each macropixel bounded within [n L , n H ] = [1.0,2.0] and with the metalens thickness being h = 2λ.The green circles in Fig. 2(a) show the highest ⟨I a ⟩ a = ⟨SR • T ⟩ among 100 optimizations with different initial guesses; they agree strikingly well with N eff /N in and exhibit the predicted local maximum at D in = D opt in , indicating that this efficiency bound of Ref. [10] is both valid and tight.
Note that N eff /N in is the predicted maximal ⟨T ⟩ with the Strehl ratio fixed at SR a = 1 for all incident angles, while the inverse design here allows SR a to vary and optimizes the worst-case SR • T .Therefore, the optimized ⟨SR • T ⟩ should only be compared to N eff /N in when the optimized Strehl ratio is high.For small values of D in here, none of the optimizations reached a high Strehl ratio, presumably because incident light in the overly small aperture D in cannot spread enough to cover the much larger D out to yield the ideal output.
Next, we consider two bounds on the device thickness [15,16].In Ref. [15], Li and Hsu used the lateral spreading ∆W of spatially localized inputs to quantify the degree of nonlocality, bounding the thickness through an empirical relation h ≥ h Li&Hsu min = ∆W .In Ref. [16], Miller considered a transverse aperture that divides the system and used the number C of channels that cross the transverse aperture to quantify the degree of nonlocality, bounding the thickness with a diffraction heuristics Here, we take the "maximal internal angle" to be θ max = 90 • since an inhomogeneous refractive index profile can scatter light to all possible angles; using a smaller θ max will increase h Miller .With the lens parameters here (D out = 16λ, NA = 0.9, and FOV = 60

IV. HIGH-NA HIGH-EFFICIENCY METALENS GUIDED BY THEORETICAL BOUNDS
Having verified the theoretical bounds, we now integrate them into the design process.We still consider NA = 0.9 and FOV = 60 • , but increase the system size to D out = 50λ since parameter scanning is no longer needed.In the first step, we choose D in = D opt in = 25λ, which maximizes the efficiency bound N eff /N in .In the second step, we choose the necessary thickness.For this system size, we find h Li&Hsu min = 5λ and h Miller min = 4.5λ (Supplementary Sec.4), and we use the larger one h = h Li&Hsu min = 5λ for the inverse design.Finally, in the third step, we perform topology optimization.Here, we additionally include a regularizer in the objective function of the optimization to promote a binary design with ε r (r) being either n L or n H (Supplementary Sec. 5).We launch 100 optimizations with random initial guesses and take the best case.Supplementary Video 1 shows the evolution of the metalens structure, its Strehl ratio and transmission efficiency as a function of the incident angle, and the intensity profiles at the focal plane as the optimization progresses.Figure 3 shows the final configuration and performance.The Strehl ratio and transmission efficiency both exhibit a flat distribution across the target FOV, with ⟨T ⟩ = 98%, ⟨SR⟩ = 92%, and ⟨SR • T ⟩ = 90%.A tight, near-ideal, focus with a full-width at half maximum (FWHM) of around 0.55λ is achieved for all incident angles within the FOV [Fig.3(c-d)].Supplementary Video 2 plots the full intensity profile and field profile for all incident angles.
Integration of the theoretical bounds into the design process is crucial in enabling this optimal performance.Previous inverse designs all adopted the common choice of D in = D out . Figure 4(a-b) shows the optimized results-following identical optimization steps-when we use D in = D out = 50λ with h = λ and h = 5λ.The optimized transmission efficiency ⟨T ⟩ and Strehl ratio ⟨SR⟩ are both greatly reduced.The thickness choice is also critical.Figure 4(c) shows the optimized results when a typical thickness h = λ is used together with D in = D opt in = 25λ, which is better than with D in = D out but still fall substantially below the optimal performance of Fig. 3 where the thickness bound is adopted.
Given the non-convexity of the optimization problem, the optimized result depends on the initial guess, which is chosen randomly here.Supplementary Fig. S7 shows a histogram of the final ⟨SR•T ⟩ among the 100 optimization runs for the four geometries in Figs.3-4.
While the combination of Strehl ratio and transmission efficiency offers a comprehensive set of metrics, their measurement requires high-resolution large-area detectors and is not convenient.Therefore, a more commonly reported metric is the focusing efficiency, defined as the ratio between the transmitted flux within three FWHM around the focal peak and the incident flux [25,26,31,44,46].We choose not to optimize such a focusing efficiency here because doing so may encourage the optimization to expand the FWHM and capture more light within the inflated 3×FWHM, which increases the focusing efficiency under such definition but lowers the actual focusing quality.To facilitate comparison with prior designs, we evaluate the focusing efficiency of the design in Fig. 3 (whose FWHM of 0.55λ is diffractionlimited); the angular dependence is shown in Supplementary Fig. S6.The focusing efficiency here averages to 88.5% over the 60 • FOV, which is higher even compared to the normal-incidence focusing efficiency of previous narrow-FOV metalenses (see Supplementary Tab.S2).
We also evaluate the ideal focusing efficiency by taking the product of the transmission efficiency bound N eff /N in and the focusing efficiency of an ideal focus with unity transmission; such an upper bound on the focusing efficiency is shown as solid lines in Fig. 1(b) for D in = D out and D in = D opt in .The design in Fig. 3 saturates the max-imized upper bound.

V. OUTLOOK
This work shows that theoretical bounds can play an active role in photonic design, identifying optimal system parameters and maximizing the efficiency limit itself.The maximizing-upper-limit strategy here can help a wide range of photonic devices reach their highest possible performance.

FIG. 1 .
FIG. 1. (a)Schematic of a high-NA wide-field-of-view metalens, with Din (Dout), θin, and h denoting the input (output) aperture diameter, incident angle, and lens thickness.(b) Comparison of the focusing efficiency in existing wide-FOV metalens designs and in this work (purple for monochromatic, green for achromatic).Details are listed in Supplementary Tab.S1.Solid lines show the theoretical limit[10] with Din = Dout and with the optimal Din = D opt in used here.

FIG. 2 .
FIG.2.Validation of theoretical bounds on the channelaveraged total transmission efficiency T and on the device thickness h for a high-NA wide-field-of-view metalens.Black solid lines are bounds on the angle-averaged transmission efficiency ⟨T ⟩ from Ref.[10], assuming a perfect Strehl ratio SR = 1 across all incident angles within the FOV.Vertical dot-dashed lines in (b) indicate the thickness bounds from Li and Hsu[15] and from Miller[16]. Green circles are the angle-averaged ⟨SR • T ⟩ from gray-scale topology optimizations.Lens parameters: numerical aperture NA = 0.9, output aperture diameter Dout = 16λ, field of view FOV = 60 • , refractive index range =[1,2].Thickness h = 2λ in (a), and input diameter Din = D opt in = 8λ in (b).
• ), the Li & Hsu bound yields h Li&Hsu min = 1.7λ, and the Miller bound yields h Miller min = 4λ.These bounds are shown as vertical dot-dashed lines in Fig. 2(b).Note that the value of h Miller min also implicitly depends on how the channel number C is counted, as detailed in Supplementary Sec. 4. The green circles in Fig. 2(b) show the highest optimized ⟨SR • T ⟩ among 100 initial guesses with D in = D opt in = 8λ.The optimized efficiency is low when the thickness is smaller than h Li&Hsu min .When the thickness goes above h Li&Hsu min , the optimized ⟨SR • T ⟩ converges toward N eff /N in , validating both the efficiency bound and the thickness bound.In this example, h Li&Hsu min marks the transition between low-efficiency and high-efficiency designs, while h Miller min coincides with the thickness at which the efficiency bound N eff /N in is reached.

9 FIG. 3 .
FIG. 3.An inverse-designed high-NA wide-FOV metalens guided by theoretical bounds.(a) Optimized metalens structure, with color indicating the refractive index (between 1.0 and 2.0).(b) The Strehl ratio and transmission efficiency T of the optimized metalens as a function of the incident angle θin.Vertical dashed lines mark the FOV.(c) Intensity profiles |Ex(y, z)| 2 for incident angles θin = 0 • and 30 • .White dashed lines mark the boundary of the metalens.(d) Intensity profiles at the focal plane z = f for different incident angles, comparing outputs from the designed structure (solid lines) with the ideal outputs assuming perfect transmission and perfect Strehl ratio (dashed lines).Lens parameters: NA = 0.9, FOV = 60 • , Dout = 50λ, Din = D opt in = 25λ, h = h Li&Hsu min

FIG. 4 .
FIG.4.Inverse-designed metalenses without guidance by theoretical bounds.The lens parameters and optimization procedure are identical to those of Fig.3except that Din = Dout = 50λ in (a-b) and the thickness is reduced to h = λ in (a,c).