1. 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 [1–5], efficiency (e.g., reflection, transmission, absorption) [6–10], 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.

An important application that calls for further progress is designing metalenses with a high numerical aperture (NA) over a wide field of view (FOV) [Fig. 1(a)], which can be used for high-resolution imaging in microscopy, endoscopy, and lithography. Metalenses that offer diffraction-limited resolution at ${\rm NA} \ge 0.9$ have been proposed [18–28], with focusing efficiencies exceeding 70% reached through inverse design [29–31]. However, these metalenses are only designed to operate at the normal incidence, and their focusing quality deteriorates sharply at oblique incidence [32,33], severely limiting the angular FOV. There are multiple strategies to broaden the FOV while preserving diffraction-limited focusing, including using an aperture to separate different incident angles onto different parts of the metasurface [34–40], using a doublet [41–47], or stacking multiple structural layers [48]. However, there is ample room for improvement in the efficiency, as summarized in Fig. 1(b) and Supplement 1, Table S1. For ${\rm NA} \ge 0.5$, no existing design achieved a focusing efficiency above 50% over a wide FOV.

 

Fig. 1. (a) Schematic of a high-NA wide-field-of-view metalens, with ${D_{{\rm in}}}$ (${D_{{\rm out}}}$), ${\theta _{{\rm in}}}$, 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 Supplement 1, Table S1. Solid lines show the theoretical limit [10] with ${D_{{\rm in}}} = {D_{{\rm out}}}$ and with the optimal ${D_{{\rm in}}} = D_{{\rm in}}^{{\rm opt}}$ used here.

Download Full Size | PDF

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 channel-averaged transmission efficiency [10], which depends on the sizes of the input and output apertures [${D_{{\rm in}}}$ and ${D_{{\rm out}}}$ in Fig. 1(a)]. Given any output aperture ${D_{{\rm out}}}$ of interest, we identify the optimal input aperture size $D_{{\rm in}}^{{\rm opt}}$ 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 ${D_{{\rm in}}}$ and the minimal $h$. Applying this approach to a metalens with ${\rm NA} = {0.9}$ 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.

2. NUMERICAL OPTIMIZATION METHOD

We first describe our inverse design methodology. We employ free-form topology optimization [49–52], 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 $\theta _{{\rm in}}^a$ within the FOV, the desired focal spot position is $\textbf{r}_{\rm f}^a = (y = f\tan \theta _{{\rm in}}^a,z = f)$, where $f$ is the focal length. We quantify the focusing quality by the ratio between the intensity of the actual design at $\textbf{r}_{\rm f}^a$ and the focal spot intensity for an ideal lens (i.e., perfectly transmitting and free of aberrations) as

We map out the focal intensity as a function of the incident angle through the generalized transmission matrix $\textbf{t} = \textbf{C}{\textbf{A}^{- 1}}\textbf{B}$ of the metalens [33]. Here, matrix $\textbf{A} = - {\nabla ^2} - {(2\pi /\lambda)^2}{\varepsilon _{\rm r}}(\textbf{r})$ is the discretized Maxwell differential operator on ${E_x}(y,z)$ at wavelength $\lambda$ for the metalens structure defined by its relative permittivity profile ${\varepsilon _{\rm r}}(\textbf{r})$. The $a$-th column of the $M$-column input matrix $\textbf{B} = [{\textbf{B}_1}, \cdots ,{\textbf{B}_M}]$ contains the source profile that generates an incident plane wave at the $a$-th incident angle $\theta _{{\rm in}}^a$ within the input aperture. This source profile is obtained by projecting $\exp (ik_y^a y)w(y)$ onto the set of propagating plane waves $\{\exp (ik_y^{{a^\prime}}y)\}$ with $|k_y^{{a^\prime}}| \lt 2\pi /\lambda$, where $w(y)$ is a square window function within $|y| \lt {D_{{\rm in}}}/2$. The $b$-th row of the $M$-row output projection matrix $\textbf{C} = [{\textbf{C}_1}; \cdots ;{\textbf{C}_M}]$ performs angular spectrum propagation [55] that propagates ${E_x}$ on the output surface of the metalens to position $\textbf{r}_{\rm f}^b$ on the focal plane. Then, the diagonal elements ${t_{\textit{aa}}}$ are the field amplitudes ${E_x}(\textbf{r}_{\rm f}^a;\theta _{{\rm in}}^a)$ at the focal spots in Eq. (1), for all incident angles $\{\theta _{{\rm in}}^a\}$ within the FOV of interest. More details are given in Supplement 1, Sections 1 and 2.

To reach the optimal ${I_a} = {{\rm SR}_a} \cdot {T_a}$ for all incident angles, 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 $g$:

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 $\Delta x$, the gradient of ${I_a}$ with respect to the permittivity profile ${\varepsilon _{\rm r}}$ of the metalens is calculated via the adjoint method (see Supplement 1, Section 3) as

During the optimizations, we update the permittivity profile with a macropixel size of $4\Delta x = \lambda /10$ 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 $\Delta x = \lambda /40$ for accuracy. Since the desired response is symmetric, we impose ${\varepsilon _{\rm r}}(\textbf{r})$ to be mirror-symmetric with respect to the lens center $y = 0$ and only parameterize ${\varepsilon _{\rm r}}$ of the macropixels in the left half of the metalens [red box in Fig. 3(a)] as the optimization variables [60].

3. 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 intuition-based designs are not likely to reach these bounds. For this validation, we consider a relatively small system: metalenses with ${\rm NA} = 0.9$, ${\rm FOV} = {60^ \circ}$, and output diameter ${D_{{\rm out}}} = 16\lambda$. Note that when a lens is used for imaging, the most relevant parameters are the spatial size of the focus, the working distance, and the total number of resolvable points. With diffraction-limited focusing, these parameters together specify the NA, ${D_{{\rm out}}}$, and FOV, while the size ${D_{{\rm in}}}$ of the input aperture remains a free variable.

In Ref. [10], we derived that the transmission efficiency averaged over all input channels within the FOV, $\langle T\rangle$, cannot exceed ${N_{{\rm eff}}}/{N_{{\rm in}}}$. Here, ${N_{{\rm in}}}$ is the number of input channels, and the inverse participation ratio ${N_{{\rm eff}}} = {({\sum\nolimits_i \sigma _i^2})^2}/\sum\nolimits_i \sigma _i^4$ quantifies the effective number of high-transmission channels using the singular values $\{{\sigma _i}\}$ of the transmission matrix. After writing down the ideal transmission matrix of diffraction-limited wide-FOV metalenses in air (Supplement 1, Section 2), we plot ${N_{{\rm eff}}}/{N_{{\rm in}}}$ as the black solid lines in Fig. 2. The efficiency bound ${N_{{\rm eff}}}/{N_{{\rm in}}}$ depends on the diameter ${D_{{\rm in}}}$ of the input aperture and has a local maximum at ${D_{{\rm in}}} = D_{{\rm in}}^{{\rm opt}} \approx 8\lambda$.

 

Fig. 2. Validation of theoretical bounds on the channel-averaged 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 $\langle T\rangle$ from Ref. [10], assuming a perfect Strehl ratio ${\rm SR} = 1$ across all incident angles within the FOV. Vertical dotted-dashed lines in (b) indicate the thickness bounds from Li and Hsu [15] and from Miller [16]. Green circles are the angle-averaged $\langle {\rm SR} \cdot T\rangle$ from gray-scale topology optimizations. Lens parameters: numerical aperture ${\rm NA} = {0.9}$, output aperture diameter ${D_{{\rm out}}} = 16\lambda$, field of view ${\rm FOV} = 60^\circ$, refractive index ${\rm range} = [1,2]$. Thickness $h = 2\lambda$ in (a), and input diameter ${D_{{\rm in}}} = D_{{\rm in}}^{{\rm opt}} = 8\lambda$ in (b).

Download Full Size | PDF

To verify this dependence, we perform the topology optimization of Eq. (2) for different ${D_{{\rm in}}}$, with the refractive index of each macropixel bounded within $[{n_L},{n_H}] = [1.0,2.0]$ and with the metalens thickness being $h = 2\lambda$. The green circles in Fig. 2(a) show the highest ${\langle {I_a}\rangle _a} = \langle {\rm SR} \cdot T\rangle$ among 100 optimizations with different initial guesses; they agree strikingly well with ${N_{{\rm eff}}}/{N_{{\rm in}}}$ and exhibit the predicted local maximum at ${D_{{\rm in}}} = D_{{\rm in}}^{{\rm opt}}$, indicating that this efficiency bound of Ref. [10] is both valid and tight.

Note that ${N_{{\rm eff}}}/{N_{{\rm in}}}$ is the predicted maximal $\langle T\rangle$ with the Strehl ratio fixed at ${{\rm SR}_a} = 1$ for all incident angles, while the inverse design here allows ${{\rm SR}_a}$ to vary and optimizes the worst-case ${\rm SR} \cdot T$. Therefore, the optimized $\langle {\rm SR} \cdot T\rangle$ should only be compared to ${N_{{\rm eff}}}/{N_{{\rm in}}}$ when the optimized Strehl ratio is high. For small values of ${D_{{\rm in}}}$ here, none of the optimizations reached a high Strehl ratio, presumably because incident light in the overly small aperture ${D_{{\rm in}}}$ cannot spread enough to cover the much larger ${D_{{\rm out}}}$ 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 $\Delta W$ of spatially localized inputs to quantify the degree of nonlocality, bounding the thickness through an empirical relation $h \ge h_{{\min}}^{{\rm Li\& Hsu}} = \Delta W$. 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 $h \ge h_{{\min}}^{{\rm Miller}} = (C\lambda)/[{2(1 - \cos {\theta _{{\max}}}){n_H}}]$. Here, we take the “maximal internal angle” to be ${\theta _{{\max}}} = {90^ \circ}$ since an inhomogeneous refractive index profile can scatter light to all possible angles; using a smaller ${\theta _{{\max}}}$ will increase $h_{{\min}}^{{\rm Miller}}$. With the lens parameters here (${D_{{\rm out}}} = 16\lambda$, ${\rm NA} = {0.9}$, and ${\rm FOV} = 60^\circ$), the Li and Hsu bound yields $h_{{\min}}^{{\rm Li\& Hsu}} = 1.7\lambda$, and the Miller bound yields $h_{{\min}}^{{\rm Miller}} = 4\lambda$. These bounds are shown as vertical dotted-dashed lines in Fig. 2(b). Note that the value of $h_{{\min}}^{{\rm Miller}}$ also implicitly depends on how the channel number $C$ is counted, as detailed in Supplement 1, Section 4.

The green circles in Fig. 2(b) show the highest optimized $\langle {\rm SR} \cdot T\rangle$ among 100 initial guesses with ${D_{{\rm in}}} = D_{{\rm in}}^{{\rm opt}} = 8\lambda$. The optimized efficiency is low when the thickness is smaller than $h_{{\min}}^{{\rm Li\& Hsu}}$. When the thickness goes above $h_{{\min}}^{{\rm Li\& Hsu}}$, the optimized $\langle {\rm SR} \cdot T\rangle$ converges toward ${N_{{\rm eff}}}/{N_{{\rm in}}}$, validating both the efficiency bound and the thickness bound. In this example, $h_{{\min}}^{{\rm Li\& Hsu}}$ marks the transition between low-efficiency and high-efficiency designs, while $h_{{\min}}^{{\rm Miller}}$ coincides with the thickness at which the efficiency bound ${N_{{\rm eff}}}/{N_{{\rm in}}}$ is reached.

4. 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 ${\rm NA} = {0.9}$ and ${\rm FOV} =\def\LDeqbreak{} 60^\circ$, but increase the system size to ${D_{{\rm out}}} = 50\lambda$ since parameter scanning is no longer needed. In the first step, we choose ${D_{{\rm in}}} = D_{{\rm in}}^{{\rm opt}} = 25\lambda$, which maximizes the efficiency bound ${N_{{\rm eff}}}/{N_{{\rm in}}}$. In the second step, we choose the necessary thickness. For this system size, we find $h_{{\min}}^{{\rm Li\& Hsu}} = 5\lambda$ and $h_{{\min}}^{{\rm Miller}} = 4.5\lambda$ (Supplement 1, Section 4), and we use the larger one $h = h_{{\min}}^{{\rm Li\& Hsu}} = 5\lambda$ 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 $\sqrt {{\varepsilon _{\rm r}}(\textbf{r})}$ being either ${n_L}$ or ${n_H}$ (Supplement 1, Section 5). We launch 100 optimizations with random initial guesses and take the best case.

Visualization 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 $\langle T\rangle = 98\%$, $\langle {\rm SR}\rangle = 92\%$, and $\langle {\rm SR} \cdot T\rangle = 90\%$. A tight, near-ideal, focus with a full-width at half maximum (FWHM) of around $0.55\lambda$ is achieved for all incident angles within the FOV [Figs. 3(c) and 3(d)]. Visualization 2 plots the 2D field and intensity profiles at all incident angles.

 

Fig. 3. 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) Strehl ratio SR and transmission efficiency $T$ of the optimized metalens as a function of the incident angle ${\theta _{{\rm in}}}$. Vertical dashed lines mark the FOV. (c) Intensity profiles $|{E_x}(y,z{)|^2}$ for incident angles ${\theta _{{\rm in}}} = {0^ \circ}$ and 30°. White dashed lines mark the boundary of the metalens. See Visualization 2 for the field and intensity profiles at all angles. (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: ${\rm NA} = {0.9}$, ${\rm FOV} = 60^\circ$, ${D_{{\rm out}}} = 50\lambda$, ${D_{{\rm in}}} = D_{{\rm in}}^{{\rm opt}} = 25\lambda$, $h = h_{{\min}}^{{\rm Li\& Hsu}} = 5\lambda$.

Download Full Size | PDF

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 ${D_{{\rm in}}} = {D_{{\rm out}}}$. Figures 4(a) and 4(b) show the optimized results—following identical optimization steps—when we use ${D_{{\rm in}}} = {D_{{\rm out}}} = 50\lambda$ with $h = \lambda$ and $h = 5\lambda$. The optimized transmission efficiency $\langle T\rangle$ and Strehl ratio $\langle {\rm SR}\rangle$ are both greatly reduced. The thickness choice is also critical. Figure 4(c) shows the optimized results when a typical thickness $h = \lambda$ is used together with ${D_{{\rm in}}} = D_{{\rm in}}^{{\rm opt}} = 25\lambda$, which is better than with ${D_{{\rm in}}} = {D_{{\rm out}}}$ but still falls substantially below the optimal performance of Fig. 3 where the thickness bound is adopted.

 

Fig. 4. Inverse-designed metalenses without guidance by theoretical bounds. The lens parameters and optimization procedure are identical to those of Fig. 3 except that ${D_{{\rm in}}} = {D_{{\rm out}}} = 50\lambda$ in (a), (b) and the thickness is reduced to $h = \lambda$ in (a), (c).

Download Full Size | PDF

Given the non-convexity of the optimization problem, the optimized result depends on the initial guess, which is chosen randomly here. Supplement 1, Fig. S7 shows a histogram of the final $\langle {\rm SR} \cdot T\rangle$ among the 100 optimization runs for the four geometries in Figs. 3 and 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 FWHMs 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 $3 \times$ 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 $0.55\lambda$ is diffraction-limited); the angular dependence is shown in Supplement 1, 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 Supplement 1, Table S2).

We also evaluate the ideal focusing efficiency by taking the product of the transmission efficiency bound ${N_{{\rm eff}}}/{N_{{\rm in}}}$ 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 ${D_{{\rm in}}} = {D_{{\rm out}}}$ and ${D_{{\rm in}}} = D_{{\rm in}}^{{\rm opt}}$. The design in Fig. 3 saturates the maximized upper bound.

5. 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 two bounds we employ [10,15] come from writing down the transmission matrix for the target function, so the strategy here may be applied to a wide range of multi-channel optical systems.

Two-layer systems, such as aperture-stop-based metasurfaces [34–40] and metasurface doublets [41–47], are simpler to design and fabricate. To separate light coming from different angles, such two-layer structures will require a larger distance between layers [39] than the generic thickness bound [15]. Since the efficiency bound [10] applies to any structure including layered ones, one can still use the maximizing-efficiency-bound strategy to determine the optimal aperture size. It will be interesting to explore how close layered structures can approach the general efficiency limit.

The design of full 3D structures is also an important future direction. The procedure for establishing bounds and maximizing the efficiency limit is the same as in 2D: one can still write down the target transmission matrix in 3D, just with more channels to account for the additional dimension and polarization. The computing cost of the inverse design is currently the bottleneck for large-area structures in 3D. One can reduce the computing cost by considering axisymmetric structures [48,61] and by dividing the structure into smaller segments [30,62–64].

Since we only consider one wavelength in this work, the performance of the resulting design is expectedly narrowband (Supplement 1, Fig. S8). Another direction for future work is to tackle broadband designs. To do so, one may consider an additional bandwidth bound [1–4] or adopt a multi-functional bound [65] when looking for the optimal parameters.