# Inverse Design of High-NA Metalens for Maskless Lithography

Haejun Chung  Department of Electronic Engineering, Hanyang University, Seoul,
04763, South Korea, e-mail<EMAIL_ADDRESS>Feng Zhang  LumArray, Inc.,
15 Ward Street, Somerville, MA 02143, USA  Hao Li  Department of Applied
Physics and Energy Sciences Institute, Yale University, New Haven, Connecticut
06511, USA  Owen D. Miller  Department of Applied Physics and Energy Sciences
Institute, Yale University, New Haven, Connecticut 06511, USA, e-mail:
<EMAIL_ADDRESS>Henry I. Smith  LumArray, Inc., 15 Ward Street,
Somerville, MA 02143, USA  Department of Electrical Engineering and Computer
Science, Massachusetts Institute of Technology (MIT), Cambridge, MA 02139 USA,
e-mail<EMAIL_ADDRESS>
###### Abstract

We demonstrate an axisymmetric inverse-designed metalens to improve the
performance of zone-plate-array lithography (ZPAL), one of the maskless
lithography approaches, that offer a new paradigm for nanoscale research and
industry. First, we derive a computational upper bound for a unit-cell-based
axisymmetric metalens. Then, we demonstrate a fabrication-compatible inverse-
designed metalens with 85.50% transmission normalized focusing efficiency at
0.6 numerical aperture at 405nm wavelength; a higher efficiency than a
theoretical gradient index lens design (79.98%). We also demonstrate
experimental validation for our axisymmetric inverse-designed metalens via
electron beam lithography. Metalens-based maskless lithography may open a new
way of achieving low-cost, large-area nanofabrication.

##  I Introduction

Zone-plate-array lithography (ZPAL) creates patterns of arbitrary geometry by
overlapping focal spots produced by an array of microlenses [1, 2, 3]. Such
maskless lithography avoids the cost and delay time associated with procuring
a photomask [4]. It also avoids troublesome coherence problems associated with
mask-based projection photolithography. Figure 1(a) is a schematic of the ZPAL
technology. A significant obstacle for ZPAL is the need for a high-efficiency,
high-numerical-aperture (NA) microlenses. Recently, the advent of the
”metalens” has led to the development of structured, wavelength-scale devices
offering superior performance to conventional approaches [5, 6, 7, 8, 9, 10].
A standard metalens design approach, called unit-cell design, stitches
subwavelength resonators together into a larger device, demonstrating the
possibility of focusing [6, 7, 8, 9, 10], but it has suffered from low
focusing efficiencies or low-NA restrictions. Recently, it has been shown that
the unit-cell design approach cannot have high efficiency at high-NA due to a
fast oscillation of the field phase and amplitude profiles and a local
periodicity violation [11].

In this work, we apply inverse design, a large-scale computational technique
[12, 13, 14, 15, 16, 17], to discover a high-NA metalens at 405nm wavelength
to realize a high-resolution, high-efficiency ZPAL lithography system. Inverse
design enables fast computation of Figure-of-Merit (FoM) gradients with
respect to design parameters over the entire design space. Fabrication-
constraints are implemented by penalization technique [18, 19] and constrained
optimization [20, 21], enabling fast discovery of fabrication-compatible
designs.

##  II Method

###  II.1 Zone-plate-array lithography: The microlenses

Figure 1: (a) Schematic depiction of zone-plate-array lithography (ZPAL). A CW
laser illuminates a spatial-light modulator, each pixel of which controls the
light intensity to one zone-plate of the array. By adjusting the focal-spot
intensity from zero to maximum in a quasi-continuous manner, linewidth can be
controlled and proximity-effects corrected. By moving the stage under computer
control, while intelligently modulating focal-spot intensities, patterns of
arbitrary geometry can be written. (b) Schematic depiction of forward
simulation of inverse design. A circularly-polarized plane wave is excited
from the substrate. $E^{dir}_{\rho}(x_{0}),E^{dir}_{\phi}(x_{0})$ at the focal
point are stored for the adjoint simulation. (c) A conjugated
$E^{dir}_{\rho}(x_{0}),E^{dir}_{\phi}(x_{0})$ field now back-propagates to
obtain adjoint electric fields at the designable region ($x^{\prime}$).

The original implementation of ZPAL utilized binary $\pi$-phase-shifting zone
plates formed in a spin-on glass with index of refraction of n $=$ 1.47. It is
well known that binary $\pi$-phase gratings [22], in which the zeroth order is
canceled and second and third orders are cut off, have a diffraction
efficiency of $\sim$42% into the $+1$-order [23]. Based on this, one would
expect a maximum focal efficiency of about 40% for a binary $\pi$-phase-
shifting zone plate [23]. Assuming this, 60% of the transmitted light would
constitute background exposure. This background is composed of residual zeroth
order as well as a -1-order virtual focus and third order foci, both real and
virtual. Because ZPAL’s writing scheme consists of overlapping non-coherent
focal spots, interference effect are avoided and linear superposition applies.
As a consequence, proximity-effect correction (PEC) can be easily implemented
if the focal spot’s point-spread function, including the broad background, are
known. Although PEC can compensate for a high background level (as is done in
scanning-electron-beam lithography where electron backscattering produces a
large, broad background) it nevertheless is highly desirable to reduce this
background.

In projection photolithography, it is common to express the minimum practical
linewidth achievable in dense patterns as,

$\textrm{W}_{\textrm{min}}=\textrm{k}_{1}\lambda/\textrm{NA}$ (1)

where $\lambda$ is the optical wavelength, NA is the len’s numerical aperture
and $\textrm{k}_{1}$ is a proportionality factor, obtained empirically. In
practice, the lower the background level, the lower one can push
$\textrm{k}_{1}$ in pursuit of finer linewidths.

In seeking microlenses with higher focal efficiencies than $\pi$-phase zone
plates, the obvious first approach would be shaping the individual zones, to
produce a gradual phase delay across them, a scheme referred to as “blazing”
in the case of gratings [24, 25]. Such a microlens is commonly called a
Fresnel lens [26, 27]. Aside from the difficulty of performing such blazing
with existing nanofabrication tools, it is well known that the focal
efficiency of Fresnel lenses drops rapidly with increasing NA [8]. Some
improvement can be achieved by so-called effective-index modulation (EIM) [28,
29] in which the thickness of the dielectric is uniform from zone to zone, but
across each zone dielectric material is selectively removed to achieve a
desired variation of phase delay. Not surprisingly this approach also
encounters decreasing focal efficiency with increasing NA [11]. Our intuitive
models about how light behaves in transiting and emerging from such complex
structures break down when diffraction angles get large.

In recent years, several groups have investigated so-called metasurfaces which
includes metalenses [11, 6, 30, 7, 9, 31, 32]. The term metasurface was
introduced to convey the notion of metal or dielectric structures in which the
dimensions of features are below the wavelength of light. In many cases, one
can think of these small structures behaving as sub-wavelength antennas.

High-Numerical-Aperture (NA) metalenses offer the possibility of significantly
improving the resolution of the ZPAL maskless lithography system. Here, we
study the theoretical efficiency limits of high-NA metalenses and then use
inverse design to realize a high-efficiency high-NA metalens. Our inverse-
designed metalens shows 85% efficiency, close to the theoretical limit. We
also fabricated metalens on SiO2 substrates to demonstrate the feasibility of
our approach.

###  II.2 Upper bound of unit-cell design

Figure 2: (a) The exit fields of a unit cell design (red) cannot exactly match
those of the ideal focusing metalens (blue). (b,c) These field differences at
the exit planes lead to reduced intensities at the focal point, as plotted
along both transverse directions.

To start, we ask: what is the largest efficiency a metalens designed by the
unit-cell method could achieve? We apply an analytical technique first
developed by some of the authors in Ref. [11]. The idea is as follows. In the
unit-cell method, one approximates the response of each metalens unit as
though it were part of a periodic structure. The collective plane of outgoing
fields should then comprise the stitched-together fields from each individual
unit cell. Yet even at this simple level of construction, one can already
identify a source of error that must emerge. The stitched-together fields
provably are not exact solutions of Maxwell’s equations. The most favorable
assumption is to take the outgoing fields to be the closest projection of the
stitched-together fields onto a complete Maxwell basis. Once one makes this
step, however, then it becomes clear that some efficiency must have been lost,
as the projected fields will not propagate to the focal point identically to
how the original fields would have. The resulting efficiency is an upper bound
to the focusing efficiency of a unit-cell design. The mathematical derivations
of the various statements above is given in Ref. [11]. We extend the 2D
computations from that paper to 3D computations that are shown in Fig. 2. The
exit plane fields show how the optimal stitched-together (“ideal”) fields are
modified, just moderately, into the nearest projection of Maxwell fields.
These differences can be more visibly significant in the focal plane, where,
for parameters matching those of the later metalenses (NA = 0.6, unit-cell
period = 0.42$\lambda$), the maximum focusing efficiency of a unit-cell design
is about 63%.

###  II.3 Inverse design of metalenses

Figure 3: Plot of the Figure-of-Merit (FoM), the sum of derivatives of FoM
with respect to electric fields
($\sum|\frac{\partial\textrm{FoM}}{\partial\mathbf{E}(\mathbf{r})}|$), and the
penalization factors ($\sum|\xi(\mathbf{r})|$) versus inverse design
iterations. For the first 50 iterations, derivatives of FoM dominate over
penalization factors, which means that the adjoint derivatives are the main
driving factors for updating design parameters. After the 50${}_{\textrm{th}}$
iteration, penalization factors gradually dominate the adjoint derivatives,
making the design parameters converge to the binary level.

Inverse design, with “adjoint”-based optimization, computes gradients with
respect to all structural degrees of freedom within forward and backward
simulations [12, 11, 33, 14]. Its origin can be traced back to circuit theory,
control theory [34], quantum dynamics [35], and deep learning [36, 37, 38]. It
has been used in nanophotonics [39, 12, 13] for emerging applications such as
tunable metasurfaces [16], solar cells [40], waveguide demultiplexers [13,
39], photolithographic mask [41], and CMOS image sensors [42, 43]. Inverse
design requires definition of a FoM and its design parameters. In this work,
we confine our design space to axisymmetric geometries for faster
computations; we can expect this assumption to have little to no effect on the
ultimate focusing efficiency, since the ideal focusing functionality can be
expected to exhibit axial symmetry as well. We use the full-wave Finite
Difference Time Domain (FDTD) method [44] to avoid assumptions, potentially
ending up with focusing efficiency drops in unit-cell approaches [11], in the
design procedure. For the FOM, we use intensity maximization at the focal
point, which generally corresponds to the maximization of focusing efficiency
in the far field [45, 46]. Our FoM is given by

$\mathcal{F}=|\mathbf{E}(\textrm{x}_{0})|^{2}$ (2)

where $\textrm{x}_{0}$ is a focal spot at the given lens dimension and NA, as
shown in Fig. 1(b), (c). Required computations can be further optimized
through near-field to far-field transformation [14] or planewave order
decomposition [11]; however, the benefits of applying these techniques are
marginal in high-NA metalenses due to a short focal length. Each geometrical
parameter is a nano-ring with 40nm width, 860nm height. The radius of a nano-
ring is defined as $40\cdot\textrm{N}$ nm (N is the index of geometry
parameter). Our metalens design consists of a SiO2 substrate (measured
refractive index of 1.47 at 405 nm wavelength), 100-nm-thick ITO (measured
refractive index of 2.10 at 405 nm wavelength), and 680-nm-thick electron-beam
resist CSAR-62 (measured refractive index of 1.59 at 405 nm wavelength). For
the geometrical degrees of freedom of nano-rings, we confine the density of
CSAR-62 between 0 and 1 at every nano-ring (40-nm-width, 680-nm-height), and
then add penalization functions [18, 19] to the FoM in Eq. 2 to ensure a
binary-material constraint. A circularly- polarized plane wave is excited at
the substrate, and then $E^{dir}_{\rho}(x_{0}),E^{dir}_{\phi}(x_{0})$ at the
focal point are calculated for the adjoint simulation. A conjugated
$E^{dir}_{\rho}(x_{0}),E^{dir}_{\phi}(x_{0})$ field back-propagates with an
adjoint source given by

$\mathbf{J}_{\rm adj}=-i\omega\mathbf{P}_{\rm
adj}=-i\omega\partial\mathcal{F}/\partial\mathbf{E}$ (3)

where $\mathbf{P}$ is an source dipole density [12]. At the design space,
derivatives of the FoM can be calculated via
$\partial\mathcal{F}/\partial\varepsilon(\mathbf{x})=\real\left[\mathbf{E}_{\rm
dir}(\mathbf{x})\cdot\mathbf{E}_{\rm adj}(\mathbf{x})\right]$, where
$\mathbf{E}_{\rm dir}$ is a direct electric field calculated via the forward
simulation and $\mathbf{E}_{\rm adj}$ is an electric field obtained via the
adjoint simulation [11, 12]. The transmission normalized focusing efficiency,
in this work, is defined as

$\eta=\frac{{\rm P}_{\rm 3rd,zero}}{{\rm P}_{\rm trans}}$ (4)

where ${\rm P}_{\rm 3rd,zero}$ is the electromagnetic power within the third
zeroes of the Airy disk area while ${\rm P}_{\rm trans}$ denotes a transmitted
electromagnetic power at the metalens surface. This definition of the
transmission normalized focusing efficiency is different than that in an
imaging application where focusing efficiency is generally defined as
$\eta=\frac{{\rm P}}{{\rm P}_{\rm inc}}$. This difference arises from the
nature of the lithography application, where the incidence power can be
increased easily; thus, we can ignore reflected power, unlike imaging
applications where the incoming light is relevant.

With our approach, a combination of forward and backward simulations takes
approximately 330 seconds, including pre, and post-processing of adjoint
derivatives on 64 cores in our workstation (AMD Ryzen Threadripper PRO 3995WX,
2.7GHz processors). The FoM rapidly increases then converges in about the 25th
iteration, after which the penalization [18, 11] transforms the grayscale
refractive indices to binary values (i.e., air or CSAR-62), which takes around
another 120 iterations as shown in Fig. 3. The averaged penalization factor
over the designable region ($\sum|\xi(\mathbf{r})|$) is set to a relative
value compared to the sum of derivatives of FoM with respect to electric
fields ($\sum|\frac{\partial\textrm{FoM}}{\partial\mathbf{E}(\mathbf{r})}|$).
By taking relative penalization factors into account, the design parameters
can be updated more systematically (e.g., adjoint derivatives dominant: 1st
$\sim$ 40th iterations, penalization factor dominant: 40th $\sim$ 150th
iterations). The penalization process can be further tuned with Gaussian
filters [47], density filters [48, 49] or level-set methods [50, 51].

##  III Result

Figure 4: Effective-Index Modulation (EIM)-based theoretical lens design. (a)
Top view of the EIM-based lens for NA = 0.57 (f=97$\mu$m, d=135$\mu$m). (b)
Focal spot intensity profile in the transverse (xy) plane at the focal
distance of f = 97 $\mu$m. (c) A transverse cut of the focal spot intensity
profile in simulation. The full width at half intensity maximum (FWHM) of the
spot is 0.571 $\mu$m. (d) $\rho,z$ side view (zoomed) of the EIM-lens.
Thickness is 680nm, covering 2-$\pi$ phase change with a maximum refractive
variation of 1.0 to 1.59. Each pixel occupies 40nm radial space in cylindrical
symmetry, corresponding to our fabrication resolution. (e) Longitudinal
($\rho$,z) intensity profile where transmission efficiency and transmission
normalized focusing efficiency are 81.63%, 79.98%, respectively. (f) SEM of
the EIM metalens which experimentally had a focal efficiency of about 48%. (g)
SEM of the outer zones of the EIM metalens.

In this section, we design and fabricate effective-medium-based and inverse-
designed metalenses. Both approaches assume that we can have control within a
fabrication constraint of 40nm (width) $\times$ 680nm (height) on the size of
nano-ring. To explore a computational upper limit of the given design problem,
the Effective Index Modulation (EIM) approach can have a material degree of
freedom in grayscale (gradient refractive index) as shown in Fig. 4(a). The
inverse design only allows a refractive index of 1.59 (CSAR-62) and 1.00
(air). The refractive indices of each nano-ring were chosen by matching a
required phase profile
$\Delta\phi(r)=\frac{2\pi}{\lambda}(f-\sqrt{r^{2}+f^{2}})$ and transmitted
phase $\phi_{\rm trans.}(r)=\frac{2\pi{\rm n}(r){\rm h}}{\lambda}$ of the
gradient index material, where $\lambda$ is the design wavelength (405nm), r
is the coordinate of each nano-ring, f is the focal length (97 um), and ${\rm
n}(r)$ is a refractive index at the radius r. The minimum grid spacing of the
simulations was enforced to 40nm to satisfy fabrication constraints, and then
the final design was validated with a finer grid spacing to guarantee the
convergence of the simulation. A circularly-polarized planewave was excited at
the SiO2 substrate, as shown in Fig. 1(b). The incident power was calculated
and then compared against a transmitted power at the plane 2$\lambda$ above
the metalens to compute transmission efficiency shown in Fig. 4(e)
(transmission normalized focusing efficiency is defined in Eq. 4). We also
verify robustness of our final design by simulating fabrication imperfections,
and maintain relatively high efficiency for moderate fabrication imperfections
as shown in Fig. 6 (Appendix).

###  III.1 Effective Index Modulated (EIM) Metalens

Figure 5: Inverse-designed metalens. (a) Top view of the inverse-designed
metalens for NA = 0.60 (f=89$\mu$m, d=135$\mu$m). (b) Focal spot intensity
profile in the transverse (xy) plane at the focal distance of f = 89 $\mu$m.
(c) A transverse cut of the focal spot intensity profile in simulation. The
full width at half intensity maximum (FWHM) of the spot is 0.611$\mu$m. (d)
zoomed $\rho,z$ side view of the metalens. Thickness is 680nm, and the minimum
design/fabrication resolution is 40nm in the radial direction with a material
refractive index of 1.59. (e) Longitudinal ($\rho$,z) intensity profile where
transmission efficiency and transmission normalized focusing efficiency are
87.46%, 85.50%, respectively. (f) Top view of the SEM image for a fabricated
inverse-designed metalens. (g) Close-up image of the SEM, showing tiny bridges
to support the high-aspect-ratio (680/40$\approx$17) nanoring-structure.

We designed metalenses for the ZPAL system at 0.57 NA with 135 $\mu$m diameter
(97 $\mu$m focal length). Figure 4(a) shows the optimized metalens with the
EIM approach. It has a refractive index gradient varying from 1.00 to 1.59.
The effective-medium-based metalens shows 81.63 % efficiency at 97 $\mu$m
focal plane while its transmission normalized focusing efficiency is 79.98 %.
Compared to the diffraction-limited focusing efficiency (total power within
the third zero of the Airy pattern $\approx$ 93.8 % [52]), our EIM-based
metalens shows relatively high efficiency even with rough fabrication
constraints (40nm $\times$ 680nm minimum feature size) as shown in Fig. 4(d).
Figures 4(f, g) show a realized EIM-based metalens where we further simplify
the metalens into 170 nm wide unit cells, resulting in a certain efficiency
drop due to the change of the unit-cell width from 40nm to 170nm. Then, we
convert grayscale refractive indices to a combination of rectangular holes
(air) and surrounding materials (CSAR-62), i.e., EIM. Rectangular holes were
programmed into each such cell, as illustrated in Fig. 4(f), with their
azimuthal angle rotated in accordance with the cell azimuth. Rotational EIM-
based metalenses were fabricated on borosilicate-glass samples (obtained from
an online vendor) coated with 185 nm of Indium-tin-Oxide (ITO). The substrates
were cleaned using UV ozone and an adhesion promoter spun on at 4000 RPM.
680-nm-thick CSAR resist was achieved at 1250 RPM. Substrates were baked on
the hotplate at 150C for 2 min. Finally, the water-soluble conducting film was
spun at 4000 RMP. Beamer proximity-effect-correction (PEC) software was used
to pattern at 1nA beam current, 2.5 nm stepping distance, 125keV beam energy,
and 15ns dwell time. This yields a base dose of 240$\mu$C/cm2.

The fabricated metalens shows transmission normalized focusing efficiency of
48 %. We assume that the conversion from 40-nm-width unit-cells to 170-nm
causes a significant efficiency drop as expected in our previous work [11].
The fabrication imperfection were relatively mild compared to the inverse-
designed metalens, discussed in the next section. As in many other
publications [6, 7, 9, 53], it is hard to validate unit-cells-based metalens
design in full-wave simulations due to a computational expense of a large area
metalens. In the next section, we fully utilize both the accuracy of the full-
wave simulation and the fast convergence of the inverse design algorithm to
discover a ‘full-wave validated’ inverse-designed metalens.

###  III.2 Inverse-designed metalens

In this subsection we optimize and fabricate an axisymmetric inverse-designed
metalens for ZPAL. To avoid the efficiency drops on: (1) the unit-cell
conversions (40-nm-width to 170-nm-width) and (2) gradient refractive indices
to EIM structure conversions, we choose two material system (air and CSAR-62),
and fabrication-compatible minimum feature size (40nm $\times$ 680nm). In this
way, the result of the full-wave simulation does not suffer efficiency drops
on such conversions. Fabrication perfection is required to maintain the
simulated efficiency.

As shown in Figs. 5(d), the inverse-designed axisymmetric metalens only has
refractive index values of 1.0 and 1.59 at 405 nm wavelength. Figure 5(e)
shows longitudinal ($\rho$,z) intensity profile where transmission efficiency
and transmission normalized focusing efficiency are 87.46%, 85.50%,
respectively. Surprisingly, the transmission normalized focusing efficiency of
our inverse-designed metalens exceeds that of the gradient refractive indices
lens by 5.2% with only a single material. We analyze the main reasons for high
efficiency in two ways. First, unit-cell-based design approaches are
relatively vulnerable to rapidly oscillating phase and amplitude profiles
because the rapid change of phase/amplitude between neighboring unit cells may
fail to create a desired continuous wavefront [11]. In the worst case
scenario, only one or two unit cells are placed to model continuous variation
of 2-$\pi$ phase shift in our 0.57 NA EIM-based metalens design. Second, the
inverse design technique inherently converges to the local optimum quickly.
Therefore, inverse design may often get stuck on bad local optima because of
complicated optimization issues [54]. However, our design problem, 0.6 NA
metalens with 135 $\mu$m diameter, has relatively strict fabrication
constraints leading to simple optimization problems in local areas. At the
same time, it is still a large-area optimization with a total of 6750 design
parameters. In other words, the design parameters are strongly correlated in
local areas but nearly uncorrelated among largely-spaced parameters.
Therefore, the summation of the local solutions found by our inverse design
could be close to the upper limit.

To fabricate an inverse-designed metalens, a dose of 194 $\mu$C/cm2 was
applied, and supporting bridges were added to avoid collapsing the high-
aspect-ratio nano-ring structure. However, despite nano-bridge support, some
material adhered to the substrate surface while parts of the upper layers were
torn away due to tensile stress. The measured transmission normalized focusing
efficiency dropped to 36%. It could be improved by changing the material to a
high index material, which reduces the aspect ratio to cover 2$\pi$ phase
range.

##  IV Conclusion

We designed a full-wave validated axisymmetric metalens for the ZPAL system
via inverse design which shows greater focusing efficiency than gradient
indices theoretical lenses. The normalized transmission efficiency of the
optimized metalens (85.50 %) is close to a theoretical upper limit for the
given dimension, meaning that axisymmetric metalens designs can have higher
efficiency than conventional convex microlens and unit-cell designed metalens.
ZPAL with metalens-array implementation may open a new way of fabricating
micro/nano-pattern at a faster and lower cost compared to nano-patterning
electron beam lithography which often suffers high-cost and slow writing time.
Furthermore, our design approach can be extended to much larger metalens
designs since both the full-wave Maxwell solver (Meep) [44] and adjoint
optimization algorithm used in this work scale well with an increasing number
of computing cores. Emerging fast-solver techniques such as integral equation-
based Maxwell solver combined with adjoint optimization can also scale this
approach to macroscopic length scales. Adjoint optimization itself has already
proven its scalability in aerodynamic design [55], where more than a billion
voxels were simulated to find optimal distributions.

##  V conflict of interest statement

The authors declare no conflicts of interest regarding this article.

##  Appendix A Robustness of the optimized metalens

Figure 6: Robustness test with (a) dilated structures, (b) eroded structures,
and (c) random variations in the optimized metalens shown in Fig 5. The
permittivities of Yee’s pixels were modified to introduce relatively small
structural variation. The permittivity change effectively represents eroded or
dilated structures via subpixel smoothing effect [44]. A ten percent change in
the permittivity degrades the transmission normalized focusing efficiency by 5
to 10 percent. It implies that the optimized metalens maintains its efficiency
over slight fabrication error.

The robustness of the optimized metalens was tested by adjusting the
permittivity of the finite pixels. The fixed pixel size makes it relatively
hard to model a slightly delated or eroded structure in the finite difference
scheme. Therefore, we utilize grayscale permittivity to model slightly delated
or eroded structures. This method was first used in the subpixel smoothing in
the FDTD [44] where interpolating the permittivity over the material surface
can significantly improve the stair-case error of the FDTD. As shown in Fig.
6, our optimized metalens has relatively robust for the slightly delated or
eroded structures.

##  Appendix B Optimized metalens data

Geometry parameters for the inverse-designed metalens is listed in this
section. Each grid occupies a nanoring of the 40-nm-width, 680-nm-height and
the radius corresponding to $40\cdot\textrm{N}$ nm (N is the index of geometry
parameter). The value “1” means permittivity of CSAR-64 at 405 nm, while the
value “0” means air:

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 0 0 0 1 1 0
0 0 0 0 0 1 0 0 1 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 1 1 0 1 1 1 1 1 0 0 1 1 1 0 0 1 1 0 1 0 1 1 1 0
0 1 1 1 1 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0
0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 0 1 1 1 0 1 1 1 0 0 1 1
1 0 1 1 0 1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0
0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 0 0 1 0 1 1 1 1 0 0 0 1 1 1 1 0 0
0 1 1 1 0 0 0 0 0 1 1 0 1 0 0 0 0 0 1 0 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 1 1
1 1 1 1 1 0 0 1 1 1 1 0 1 0 1 1 1 1 0 0 1 0 1 1 0 0 0 0 1 1 0 0 1 0 0 0 1 1 0
0 0 0 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 0 1 0 1 1 0 1 1 1 0 0 0 0 1 1 0 0
1 0 1 0 0 0 1 0 0 1 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 0 0 1 1 1 1 0 0
1 0 0 1 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 0 1 1 1 1 0 1 0 1
1 1 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1
0 1 1 0 0 1 1 1 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 1 1 1 1 0 1 1
0 0 0 0 1 1 0 1 0 0 0 0 0 1 0 1 1 0 0 0 0 1 1 1 1 1 1 1 0 0 1 1 1 1 1 0 0 0 1
1 1 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 0 1 0 1 1 1 0
0 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 1 1 1 1 0 1 1 0 1 1 1 0 1 0 1 1 1 0 0 0 0 1 0
1 1 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 0 1 1 1 0 0 0 1 1 0 1 0 1 0 0 0 1 0 0 0 1 1
1 1 1 1 1 0 0 1 1 1 1 0 0 0 1 1 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 1 1 1 1 0 0 0 1
1 1 1 0 0 1 1 1 0 0 0 0 0 1 1 0 0 0 1 1 1 1 1 1 1 0 0 1 1 0 1 1 0 0 1 1 0 0 0
1 1 0 0 0 0 0 1 1 1 1 1 1 1 0 0 1 1 1 0 1 0 0 0 1 0 0 1 1 0 0 0 0 0 0 1 1 1 1
1 1 1 0 1 1 1 0 1 0 1 1 0 0 1 0 1 1 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 0 1 0 0 1
1 0 0 1 1 0 0 0 0 0 1 1 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 1 1 0 0 0 0 1 1 1
1 1 1 0 0 1 1 1 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 1 0 0 1 0 1
0 1 0 1 0 0 0 1 1 1 1 1 1 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0
1 0 1 1 0 1 1 0 0 0 1 0 1 0 0 0 0 1 1 1 1 1 1 0 1 0 1 1 1 0 0 1 0 0 1 1 0 0 0
0 1 1 1 1 1 1 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 0 0 0 0 1
1 0 1 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 1 1 1 1 1 1 0 0 1 1
1 0 0 0 0 1 1 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 0 0 1 1 0 1 0 0 0 0 1 1 1 1 1 1
0 1 1 1 0 0 0 0 1 1 0 0 0 0 0 1 1 1 1 1 1 0 1 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 1
1 1 1 0 1 1 1 0 0 1 0 1 0 0 0 0 0 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 0 0 0 1 0 0 1
1 1 1 1 1 0 1 1 1 0 0 1 0 1 0 0 1 0 0 1 1 1 1 1 1 0 1 1 1 0 0 0 1 1 0 0 1 0 0
1 1 1 1 1 1 0 1 1 1 0 1 0 1 0 1 0 0 0 0 1 1 1 1 1 0 1 0 1 1 0 1 1 0 0 0 0 0 0
0 1 1 1 1 1 0 0 1 1 0 1 1 0 0 0 1 0 0 0 0 1 1 1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 0
0 1 1 1 1 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 1 1 1 1 0 0 1 0 1 1 0 0 0 0 0 1 1 1
1 1 1 0 1 1 1 0 0 0 1 1 0 0 0 0 0 0 1 1 1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 1
1 0 1 1 0 0 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0
0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 1
1 1 0 0 0 0 0 1 1 1 1 1 1 0 0 1 0 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0 1 0 1 1 0 0 0
0 0 0 1 1 1 1 1 1 0 1 0 1 1 0 0 0 0 0 1 1 1 1 1 1 0 1 0 1 1 1 0 0 0 0 0 0 0 0
1 1 1 1 1 0 0 1 0 1 1

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