# Imaging properties of large field-of-view quadratic metalenses and their
applications to fingerprint detection

Emmanuel Lassalle  Tobias W. W. Mass  Damien Eschimese  Anton V. Baranikov
Egor Khaidarov  Shiqiang Li  Ramon Paniagua-Dominguez
<EMAIL_ADDRESS>Arseniy I. Kuznetsov
<EMAIL_ADDRESS>Institute of Materials Research and
Engineering, A*STAR (Agency for Science, Technology and Research), 138634,
Singapore

###### Abstract

Dielectric metasurfaces, extremely thin nanostructured dielectric surfaces,
hold promise to replace conventional refractive optics, such as lenses, due to
their high performance and compactness. However, designing large field-of-view
(FOV) metalenses, which are of particular importance when imaging relatively
big objects at short distances, remains one of the most critical challenges.
Recently, metalenses implementing a quadratic phase profile have been put
forward to solve this problem with a single element, but despite their
theoretical ability to provide $180^{\circ}\,$FOV, imaging over very large FOV
has not been demonstrated yet. In this work, we provide an in-depth analysis
of the imaging properties of quadratic metalenses and, in particular, show
that due to their intrinsic barrel distortion or fish-eye effect, there is a
fundamental trade-off between the FOV achievable in a given imaging
configuration and the optical resolution of the metalens and/or the detector
resolution. To illustrate how to harness the full potential of quadratic
metalenses, we apply these considerations to the fingerprint detection
problem, and demonstrate experimentally the full imaging of a $5\,$mm
fingerprint with features of the order of $100\,\mu$m, with a metalens ten
times smaller in size and located at a distance of only $2.5\,$mm away from
the object. This constitutes the most compact system reported so far for the
fingerprint detection.

###### keywords:

Flat-optics; Metasurfaces; Metalenses; Imaging; Spherical aberrations; Barrel
distortion

These authors made equal contributions to this work. These authors made equal
contributions to this work. These authors made equal contributions to this
work. These authors made equal contributions to this work.

## Introduction

Modern electronic devices require increasing variety and number of
functionalities while demanding ever more compact and lightweight designs.
Miniaturization of integrated systems fuel research and development of new
concepts in _e.g._ data storage, sensing and optics. In recent years, flat
optics emerged as a promising platform in which conventional refractive
optical components are replaced by so called metasurfaces 1, 2, 3.
Metasurfaces are planar arrays consisting of numerous tailored nanostructures
to orchestrate subwavelength light manipulation on a macroscopic scale. They
can realize functionalities or devices such as _e.g._ polarization-sensitivity
4, 5, 6, waveplates 7, 8, holograms 9, 10, 11, 12, 13, structural color
printing 14, 15, 16, 17, 18 and metalenses 19, 20, 21, 22. The latter are the
flat optics analogue to conventional, bulk refractive lenses or lens systems,
and can be tailored to address challenges related to _e.g._ polarization-
independent imaging 23, 24, high numerical aperture (NA) focusing 25, 19,
spherical aberrations correction 26, achromatic performance 27, 28, 29, 30, 31
and wide field-of-view (FOV) imaging 32, 33, 34. In particular, conventional
wide field-of-view imaging requires lens systems consisting of multiple
comparably bulky and heavy refractive lenses 35, turning a flat optical
solution particularly attractive. In the past few years, flat optical
approaches to wide FOV imaging included _e.g._ aplanatic metasurfaces 26,
metalens doublets 36, 37 and numerically optimized phase profiles 38, 39, 40.
Generally, the difficulty in realizing wide FOV imaging using a single
metalens is to tailor simultaneously the optical response of individual
nanostructures as well as that of supercells to maintain the desired phase
profile and avoid off-axis related abberations for a wide angular range.
Recently, metalenses imparting a quadratic phase profile have been proposed to
achieve arbitrary wide FOV by trading off FOV against diffraction limited
resolution equivalent to an NA of 0.27 32, 33, 34.

Here, we provide an in-depth analysis of the imaging properties of quadratic
metalenses, identify their fundamental trade-offs and demonstrate unique
application functionality for fingerprint detection. We confirm quadratic
metalenses to be excellent flat optical solutions for wide FOV imaging but
unveil that, in practice, the effective FOV is not arbitrary large as
previously suggested 33. Our findings underline the importance of considering
barrel distortion in combination with optical resolution limit and/or detector
pixel size when designing quadratic metalenses for imaging configurations,
especially for shorter distances. After elaborating on the general imaging
properties of quadratic lenses, we demonstrate short distance imaging of a
fingerprint located in close proximity to the quadratic metalens, a
configuration suitable for portable electronic devices such as smartphones.
Our results show that, using a single quadratic metalens, we are able to image
a full fingerprint with a size of $5\,\text{mm}\times 5\,\text{mm}$ with a
total device thickness of only 2.7 mm. This is a considerable form-factor
improvement compared to previous solutions, using _e.g._ lens arrays to
address this problem 41.

## Theory of quadratic metalenses

### Focusing properties

The general behavior of a phase-gradient metasurface is governed by the so-
called “generalized Snell’s laws”, that can be derived from Fermat’s principle
42. For a metasurface working in transmission, this law states that an
incident light impinging the metasurface at an angle $\theta_{\text{i}}$ is
deflected by an angle $\theta_{\text{t}}$ given by 43:

$n_{\text{t}}\,\text{sin}\left(\theta_{\text{t}}\right)=n_{\text{i}}\,\text{sin}\left(\theta_{\text{i}}\right)+\frac{\lambda_{0}}{2\pi}\frac{\partial\varphi}{\partial
r}\;,$ (1)

where $n_{\text{i}}$ ($n_{\text{t}}$) is the refractive index of the medium
where the light is incident (transmitted), $\lambda_{0}$ is the wavelength of
light in vacuum, $\varphi$ is the phase imparted by the metasurface, and $r$
denotes the position on the metasurface, which is centered at $r=0$. This
equation can be re-written as:

$k_{\text{t}}=k_{\text{i}}+\frac{\partial\varphi}{\partial r}\;,$ (2)

where $k_{\text{i}}\equiv k_{0}\,n_{\text{i}}\,\text{sin}(\theta_{\text{i}})$
[$k_{\text{t}}\equiv k_{0}\,n_{\text{t}}\,\text{sin}(\theta_{\text{t}})$] is
the projection of the incident (transmitted) wavevector on the metasurface
plane and $k_{0}\equiv 2\pi/\lambda_{0}$ is the norm of the wavevector in
vacuum. In Eq. (2), the phase gradient $\partial\varphi/\partial r$ can be
physically interpreted “as an effective wavevector, leading to a
generalization of the conservation of the wavevector parallel to the surface”
42. A quadratic metalens is a metasurface that encodes the following quadratic
phase profile 32, 33:

$\varphi\left(r\right)=\varphi\left(0\right)-\frac{2\pi
n_{\text{t}}}{\lambda_{0}}\frac{r^{2}}{2f}\;,$ (3)

where $f$ is the focal length and the minus sign is for a converging lens.
Without loss of generality, we choose in this work the phase reference
$\varphi\left(0\right)$ to be equal to $2\pi$, but one can make any other
choice of phase reference that may possibly depend on the wavelength.
Considering that only wavevectors $\left|k_{\text{t}}\right|\leq 2\pi
n_{\text{t}}/\lambda_{0}$ propagate, this condition translates, in the case of
the quadratic phase profile, into (by plugging Eq. (3) into Eq. (2)):

$\underbrace{f\left(\frac{n_{\text{i}}}{n_{\text{t}}}\,\text{sin}\left(\theta_{\text{i}}\right)-1\right)}_{r^{\text{lower}}(\theta_{\text{i}})}\leq
r\leq\underbrace{f\left(\frac{n_{\text{i}}}{n_{\text{t}}}\,\text{sin}\left(\theta_{\text{i}}\right)+1\right)}_{r^{\text{upper}}(\theta_{\text{i}})}\;.$
(4)

This equation is illustrated in Fig. 1 (a) where we computed, based on Fourier
optics calculations (see the Methods section), the propagation of a plane wave
through a quadratic metalens for three different angles of incidence (AOI):
$\theta_{\text{i}}=0^{\circ{}}$, $30^{\circ{}}$ and $50^{\circ{}}$. One can
see in Fig. 1 (a) that the light is only transmitted through areas (in red,
green and blue, respectively) that are smaller than the physical metalens
aperture (in white). The parameters taken for the simulations are: wavelength
$\lambda_{0}=740\,$nm, focal lens $f=203\,\mu$m and physical diameter
$D=4f=812\,\mu$m, and considering air as the incident and transmitted medium
($n_{\text{i}}=n_{\text{t}}=1$). One can also see that the diameters of these
effective areas are all equal to $2f$. Moreover, these areas are
translationally shifted in the $x$-direction by the amount
$f\,\text{sin}(\theta_{\text{i}})$, as the AOI $\theta_{\text{i}}$ increases
from $0^{\circ}{}$ to $50^{\circ}{}$. Thus, as pointed out in 32, the angular
incidence is simply converted to a lateral shift of the phase profile.

We summarize hereafter the properties contained in the inequality of Eq. (4),
whom most of them have been presented in 32. First, the area that transmits
light, which we will refer to throughout this paper as the _effective_ working
area of the metalens, has a diameter defined as $D^{\text{eff}}\equiv
r^{\text{upper}}(\theta_{\text{i}})-r^{\text{lower}}(\theta_{\text{i}})$ that
reads:

$D^{\text{eff}}=2f\;.$ (5)

The previous equation implies that, even if the whole lens is illuminated,
only a portion of it is actually transmitting the incident light. This leads
to an _effective_ numerical aperture defined as $\text{NA}^{\text{eff}}\equiv
n_{\text{t}}\,R^{\text{eff}}/\sqrt{\left(R^{\text{eff}}\right)^{2}+f^{2}}$
that reads:

$\text{NA}^{\text{eff}}=n_{\text{t}}\frac{\sqrt{2}}{2}\;.$ (6)

Second, the effective working area is translationally shifted depending on the
angle of incidence by the amount
$\Delta(\theta_{\text{i}})\equiv(r^{\text{upper}}(\theta_{\text{i}})+r^{\text{lower}}(\theta_{\text{i}}))/2$
which gives:

$\Delta(\theta_{\text{i}})=\frac{n_{\text{i}}}{n_{\text{t}}}f\text{sin}(\theta_{\text{i}})\;.$
(7)

Third, by considering the maximum positive angle of incidence possible,
$\theta_{\text{i}}=90^{\circ}$, we obtain the upper position
$r^{\text{upper}}$ which reads
$r^{\text{upper}}=f\left(1+n_{\text{i}}/n_{\text{t}}\right)$. Conversely, by
considering the maximum negative angle possible,
$\theta_{\text{i}}=-90^{\circ}$, we obtain the lower position
$r^{\text{lower}}=-f\left(1+n_{\text{i}}/n_{\text{t}}\right)$. This sets the
maximum diameter of the lens that is active for the full range of angles of
incidence, what we denote as the _physical diameter_ of the metalens $D\equiv
r^{\text{upper}}-r^{\text{lower}}$, to:

$D=2f\left(1+\frac{n_{\text{i}}}{n_{\text{t}}}\right)\;,$ (8)

which leads to the corresponding physical numerical aperture:

$\text{NA}=\frac{n_{\text{t}}\left(n_{\text{i}}+n_{\text{t}}\right)}{\sqrt{n_{\text{t}}^{2}+\left(n_{\text{i}}+n_{\text{t}}\right)^{2}}}\;.$
(9)

We should stress here this important result. Namely, that there is no point in
building a quadratic metalens with a diameter (and therefore NA) larger than
the ones given by these equations.

Figure 1: Fourier optics simulations of the behaviour of a quadratic metalens
of diameter $D=812\,\mu$m and focal length $f=203\,\mu$m, at the wavelength
$\lambda_{0}=740\,$nm. (a) Cross-section in the metalens plane at $z=0$, for
three different AOI: $\theta_{\text{i}}=0^{\circ{}}$, $30^{\circ{}}$ and
$50^{\circ{}}$ in red, green and blue colours, respectively. The white area
denotes the physical aperture of the metalens, and the coloured areas
represent the effective areas for which the transmitted light is non-
evanescent. (b) Propagation of an incident plane wave in $z$-direction through
a quadratic metalens at $z=0$, same colour code. The focal plane is denoted by
the black dashed line.

(c), (d) and (e) Cross-sections of the focal spots in the plane at
$z=194\,\mu$m for the corresponding AOI, same colour code. The solid lines
denote the intensity profiles along the $x$\- and $y$\- directions passing
through the center of the focal spots. In all these color maps, the opacity
levels are proportional to the normalized intensity and for (b) the normalized
intensity is in log scale.

In Fig. 1 (b), we show the propagation of the wave coming from $z<0$ through
the quadratic metalens located in the plane $z=0$ keeping the same simulation
parameters and considering the same three different AOI:
$\theta_{\text{i}}=0^{\circ{}}$, $30^{\circ{}}$ and $50^{\circ{}}$, as
illustrated by the red, green and blue arrows, respectively. One can see in
all three cases that the light is focused around the focal plane at
$z=f=203\mu$m (denoted by the black dashed line) without apparent distortion,
demonstrating the remarkable focusing performance of such metalenses for
different AOI. Interestingly, one can also observe an elongation of the focal
spots across the $z$-direction, leading to a certain depth-of-focus (DOF),
which is the region along the $z$-direction over which the focal spot remains
tightly focused. A further analysis of the wavefront produced by this metalens
based on Zernike polynomials reveals that this elongation is mainly due to
spherical and defocus aberrations (see Supporting Information (SI) Section 1
and Figs. S1 and S2). This unveils the intrinsic non-stigmatic character of
quadratic metalenses, in contrast with an ideal lens that focuses all incoming
rays into a focal point.

The focal spot in the $xy$-plane is shown for the three cases in Figs. 1
red(c), (d) and (e), respectively. Note that the plane considered here is
where the maximum of intensity is found, at $z=194\,\mu$m, _i.e._ slightly
before the focal plane, which is an intrinsic property of this type of lenses
with spherical and defocus aberrations. One can see that in all cases the
focal spots have similar full width at half maximum (FWHM) of about $1\,\mu$m
in the $x$\- and $y$-directions. This FWHM is about two times larger than that
for a diffraction-limited lens with NA equal to $\text{NA}^{\text{eff}}\simeq
0.71$ [see Eq. (6)], which stems precisely from the spherical aberrations
mentioned above.

In order to be more quantitative, we calculated the focusing efficiency,
defined as the power within the focal spot (integrated over the second Airy
disk) divided by the power passing through the effective working area of
diameter $D^{\text{eff}}=2f$, and the FWHM of the focal spot as a function of
the AOI ranging from $\theta^{\text{min}}_{\text{i}}=0^{\circ{}}$ to
$\theta^{\text{max}}_{\text{i}}=85^{\circ{}}$. The focusing efficiency and
FWHM are found to be constant in the entire range of AOI, equal to $14\%$ and
$1\,\mu$m, respectively (see SI Section 2 and Fig. S3). Therefore, the FOV,
defined as two times the maximum AOI $\theta^{\text{max}}_{\text{i}}$ for
which the metalens is able to focus an incoming light, is extremely large, at
least equal to $\text{FOV}\geq
2\,\theta^{\text{max}}_{\text{i}}=170^{\circ}{}$. Also, the rather low
efficiency can be explained by the fact that the energy is spread within the
elongated focal spot, which is an intrinsic property of quadratic metalenses
(note that if one considered the ratio of the intensity in the focal spot
divided by the intensity that impinges the total area of the metalens of
physical diameter $D=4f$ instead, the focusing efficiency would be four times
smaller). We finally calculated the lateral displacement of the focal spot
(see SI Section 2 and Fig. S3) and found that it perfectly matches with the
analytical formula of Eq. (7), which not surprisingly shows that the lateral
displacement of the effective working area leads to a lateral displacement of
the focal spot by the same amount.

### Imaging properties

We now analyse the image formation with quadratic metalenses, based on ray-
tracing methods. We apply the _first-order_ ray-tracing method, which consists
in considering two particular rays coming from an object point to obtain the
corresponding image point as the intersection of these two rays in the image
space. Note that, as we saw in Section Focusing properties, such metalenses
are not stigmatic (_i.e._ , they do not image a point source into a single
image point) because of spherical aberrations, and therefore this approach
will only be valid in the _paraxial approximation_ (that is for object points
close to the OA and small AOI).

We thus consider two particular rays (straight red lines in Fig. 2 (a)), that
come from the same point of an object located at a distance $x_{\text{o}}$
from the OA: one ray passes the center of the metalens [labelled by $(1)$] and
is not deflected when it emerges in the image space [labelled by
$(1^{\prime})$]; the other ray is parallel to the OA in the object space
[labelled by $(2)$] and is deflected by the angle $\theta$ according to
$x_{\text{o}}=f\text{sin}\left(\theta\right)$ [Eq. (7)] in the image space
[labelled by $(2^{\prime})$]. It is this relation between the deflection angle
and the distance to the center of the lens that makes this type of lenses
behave completely different from ideal ones, for which an incident ray
parallel to the OA passes the image focal point and that lead overall to a
planar image with a certain magnification (see the SI Section 3 and Fig. S4).
In particular, it implies that the intersection of the two rays $(1^{\prime})$
and $(2^{\prime})$ in the image space for different points in the object space
obeys the following equation, in the Cartesian coordinate system of Fig. 2 (a)
(see the SI Section 3 for derivation):

$\frac{(z-z_{0})^{2}}{a^{2}_{\text{i}}}+\frac{x^{2}}{b^{2}_{\text{i}}}=1\quad\text{with}\quad\left\\{\begin{array}[]{ll}z_{0}=\frac{1}{2}\frac{d_{\text{o}}f}{d_{\text{o}}-f}\\\\[5.69054pt]
a_{\text{i}}=\frac{1}{2}\frac{d_{\text{o}}f}{d_{\text{o}}-f}\\\\[5.69054pt]
b_{\text{i}}=\frac{f^{2}}{\sqrt{2d_{\text{o}}(d_{\text{o}}-f)}}\end{array}\right.\;,$
(10)

where $f$ is the focal length and $d_{\text{o}}$ the distance between the
object and the lens (all quantities being taken as algebraic quantities). Eq.
(10) describes an ellipse centered on $(0,z_{0})$ with width $2a_{\text{i}}$
and height $2b_{\text{i}}$. Therefore, a quadratic metalens conjugates a
planar object (green line in Fig. 2 (a)) into an elliptical image (red curve
in Fig. 2 (a)). Note that this is valid for an object located at a distance
$d_{\text{o}}>f$, and for the case of an object located between the focal
plane and the lens $d_{\text{o}}<f$, the metalens creates a virtual image in
the object space that is described by an hyperbola equation (see SI Section
3). This result, which is one of the key imaging properties of this type of
lenses, can be generalized to the case of a 2D planar object: the ellipse
(hyperbola) becomes an ellipsoid (hyperboloid) with same $b_{\text{i}}$ factor
for the $x$ and $y$ coordinates.

To verify this, we computed with Fourier optics simulations in Fig. 2 (b) the
imaging by a quadratic lens of three different points of a planar object
located at a distance $d_{\text{o}}=1.5f$ from the lens. One can see that the
corresponding image points follow the ellipse given by Eq. (10) (dashed black
line in Fig. 2 (b)).

Figure 2: (a) First-order ray-tracing for a quadratic lens. A quadratic lens
(blue double-arrowed line) of focal length $f$ conjugates a planar object
located at a distance $d_{\text{o}}$ (solid green line) into an image (solid
red curve) which is formed along the ellipse given by Eq. (10) (black dashed
curve), in the two dimensional Cartesian coordinate system with origin $O$ and
axis lines $X$ and $Z$, oriented as shown by the arrows. In this
configuration, $d_{\text{o}}=3f$ and $d_{\text{i}}=1.5f$. (b) Fourier optics
simulations of the intensity maps obtained when imaging three points of a
planar object in the same configuration as in (a). Different colors correspond
to different positions of the point in the object, with opacity levels being
proportional to the normalized intensity. The ellipse given by Eq. (10) is
also shown (black dashed curve).

Another interesting result is obtained when one considers the situation in
which the distance to the object is much larger than the focal length of the
lens ($d_{\text{o}}\gg f$). In this case, usually met when imaging objects
that are located very far away from the lens, a quadratic metalens can image a
_planar_ object into a _planar_ image, which is usually what one wants to
achieve in practice. Indeed, to first order in $f/d_{\text{o}}$, the ellipse
center becomes $z_{0}\rightarrow 0.5f$ and its axis $a_{\text{i}}\rightarrow
0.5f$ so that the image is formed in the image focal plane (located at a
distance $f$ from the lens). Also, for object points very far from the OA (for
which the previous analytical approach fails), the rays will asymptotically be
parallel to each other and reach the lens with an AOI close to
$\theta_{\text{i}}=90^{\circ}{}$, and we know, following Eq. (7), that these
points will be focused in the image focal plane and at a distance $f$ from the
OA. Therefore, in the case of far-way objects ($d_{\text{o}}\gg f$), the image
created in the image focal plane is planar, and its maximum size is:

$h^{\text{max}}_{\text{i}}=2f\;,$ (11)

in stark contrast to an ideal lens. This implies that there is no point for
the sensor to have a cross-section larger than $2f\times 2f$ (unless one uses
external optics to magnify the image to match the sensor size).

The imaging properties of quadratic metalenses found in this section are
summarized in Table 1.

Object position | Image coordinates $(x,y,z)$ | Image type
---|---|---
$d_{\text{o}}<f$ | $x,y\in\left[-\infty,\infty\right]$ | Hyperboloid (3D virtual image)
$z\in\left[-\infty,-\frac{d_{\text{o}}f}{f-d_{\text{o}}}\right]$
$d_{\text{o}}=f$ | $x,y\in\left[-\infty,\infty\right]$ | Infinite
$z=\infty$
$d_{\text{o}}\gtrsim f$ | $x,y\in\left[-f,f\right]$ | Ellipsoid (3D image)
$z\in\left[f,\frac{d_{\text{o}}f}{d_{\text{o}}-f}\right]$
$d_{\text{o}}\gg f$ | $x,y\in\left[-f,f\right]$ | Planar (2D image)
$z=f$
Table 1: Coordinates of image points of an object (second column) and type of
obtained image (third column) for different regimes of distances object-lens
(first column). For an object located between the object focal plane and the
lens $d_{\text{o}}<f$, the image is virtual and formed along an hyperboloid;
for an object located in the focal plane $d_{\text{o}}=f$, the image is formed
at infinity; for distances larger but comparable to the focal lens
$d_{\text{o}}\gtrsim f$, the image is formed along an ellipsoid; for distances
much larger than the focal lens $d_{\text{o}}\gg f$, the image is formed in
the image focal plane.

## Practical implementation and characterization of quadratic metalenses

The design of phase-gradient metasurfaces is usually based on the so-called
phase-mapping approach, which consists on mapping a given phase profile with
discrete elements, called meta-atoms, which locally impart the desired phase.
Very importantly, this phase discretization also leads to some limitations of
the metalens performances. In the following, we derive such limitations with
regards to the maximum physical diameter and corresponding NA of the lens, and
also give a criterion about the FOV that be can be expected for quadratic
metalenses. We then present the experimental implementation of such lenses to
achieve an extremely large-FOV, and explain why the actual FOV exceeds by far
the one expected from theoretical considerations unveiling the very reason why
such quadratic metalenses offer such large FOV.

### Design considerations: the effect of the phase discretization

#### Maximum physical diameter and numerical aperture

In the phase-mapping approach, the different meta-atoms are enclosed into
unit-cells, and a set of unit-cells that samples the phase linearly from 0 to
$2\pi$ forms a super-cell. Let us denote by $p$ the size (or pitch) of a unit-
cell, and let us denote by $N$ the minimum number of unit-cells used to sample
the phase from 0 to $2\pi$ in the metalens ($N\geq 2$ to respect Nyquist
criterion). In other words, $Np$ is the length of the smallest super-cell of
the metalens. For a metalens with focal length $f$, the maximum physical
radius $R^{\text{max}}$ is reached when the spatial variation of the phase
$\varphi$ is equal to the ratio between $2\pi$ and $Np$:

$\left|\frac{\partial\varphi}{\partial
r}\right|_{R^{\text{max}}}=\frac{2\pi}{Np}\;.$ (12)

Therefore, for a quadratic phase profile [Eq. (3)], the maximum physical
diameter reads:

$D^{\text{max}}=\frac{2f}{n_{\text{t}}\,\xi}\quad\text{with}\quad\xi\equiv\frac{Np}{\lambda_{0}}\;.$
(13)

The parameter $\xi$ is an important figure-of-merit to characterize the
discretization of the phase profile, and we call it in the following
_discretization parameter_. One can already deduce the lower bound for this
parameter: $\xi\geq 1/(n_{\text{i}}+n_{\text{t}})$, by recalling that the
maximum reasonable physical diameter of the lens — when not limited by
discretization — is $D=2f(1+n_{\text{i}}/n_{\text{t}})$ [see Eq. (8)]. In
terms of the corresponding maximum physical NA, this leads to:

$\text{NA}^{\text{max}}=\frac{n_{\text{t}}}{\sqrt{1+\left(n_{\text{t}}\,\xi\right)^{2}}}\;.$
(14)

#### Criterion for field-of-view

This maximum physical diameter may reduce the FOV of a quadratic metalens in
practice. Indeed, by recalling that the effective working area with radius
$R_{\text{eff}}=f$ is translationally shifted as the AOI increases (according
to Eq. (7)), this area will be cut whenever the AOI exceeds a certain critical
angle $\theta_{\text{i}}^{\text{cut}}$ satisfying:

$\Delta\left(\theta_{\text{i}}^{\text{cut}}\right)+R^{\text{eff}}=R^{\text{max}}\;.$
(15)

Thus, we can define a criterion for the minimum FOV of the lens
$\text{FOV}\geq 2\,\theta_{\text{i}}^{\text{cut}}$, which, by using Eqs. (5),
(7) and (13) into Eq. (15), reads:

$\text{FOV}\geq
2\,\text{arcsin}\left(\frac{1}{n_{\text{i}}}\frac{1-n_{\text{t}}\xi}{\xi}\right)\quad\text{with}\quad\frac{1}{n_{\text{i}}+n_{\text{t}}}\leq\xi\leq\frac{1}{n_{\text{t}}}\;.$
(16)

According to this criterion, when the discretization parameter $\xi$ (or the
pitch $p$) decreases, the minimum FOV of the lens increases (see SI Section 4
and Fig. S5 for a plot of this minimum FOV). Note that this criterion does not
hold if $\xi>1/n_{\text{t}}$, since in this case
$R^{\text{max}}<R^{\text{eff}}$ and the effective working area is already cut
at normal incidence.

### Experimental realization

#### Design and fabrication

We designed and fabricated a quadratic metalens for the working wavelength
$\lambda_{0}=740\,$nm. The meta-atoms used to realize such a quadratic
metalens must, ideally, satisfy several conditions: (i) have a high-refractive
index, to ensure a good confinement of the optical fields so that the local
phase depends only on the corresponding meta-atom, _i.e._ , minimizing non-
local and inter-particle coupling effects; (ii) being lossless, to guarantee
high transmission efficiencies; and (iii) being robust, in terms of
transmission efficiency and imparted phase, against the AOI. As a good meta-
atom candidate meeting these requirements, we consider amorphous silicon (aSi)
nanopillars with circular cross-sections (see insets in Fig. 3 (a)). At
$\lambda_{0}=740\,$nm, aSi presents negligible absorption and a high
refractive index ($n=4.02$). Moreover, such nanopillars act as truncated
waveguides which, depending on their diameter, allow guided modes to propagate
with different effective refractive indices, thus imparting different phase-
delays to the light 21, 4.

Figure 3: (a) Transmission (solid red line) and phase (solid blue line) of a
periodic arrangement of identical Si nanopillars as a function of their
diameters. The left inset represents a single Si nanopillar of height $H$ and
diameter $D$ on top of a glass substrate. The right inset represents a top-
view of the hexagonal lattice with pitch $p$. A unit-cell is also highlighted
(dashed green line). The phase accumulated through propagation over a distance
$H$ in the case of an infinite circular waveguide is also shown (dashed blue
line). (b-d) SEM images of different parts of the fabricated metalens. The
scale bars represent $5\,\mu$m in (b) and (c), and $250\,$nm in (d).

In Fig. 3 (a), we simulated using the Finite-Difference Time-Domain (FDTD)
method (see Methods section) the phase imparted by a periodic array of aSi
nanopillars with height $H=350\,$nm for a normally incident wave, as a
function of the nanopillar diameters (blue line). The transmittance of the
array is also shown (red line). One can see that by varying the diameter, one
can span the whole $2\pi$-phase space while maintaining a good transmission
efficiency. We also studied the influence of the AOI of the incident wave, and
found that phases and transmittances are quite robust (see SI Section 5 and
Fig. S6). For the simulations, we considered a hexagonal lattice with pitch
$p=300\,$nm (distance between the centers of the nearest pillars), to ensure
that near-field interactions between nanopillars are negligible and that the
behavior of a single nanopillar is mostly independent from its neighbours.
This guarantees the validity of the local phase-mapping approach, which
assumes that the meta-atoms are surrounded by similar meta-atoms (local
periodicity approach), and allows to extrapolate these simulated
transmittances and phases to the metalens design.

For the design of the quadratic metalens, we therefore choose nanopillars with
height $H=350\,$nm and different diameters in the range
$D=[100\,\text{nm},220\,\text{nm}]$ in order to map the $2\pi$-wrapped phase
profile given in Eq. (3). We purposely avoid the portion
$D=[195\,\text{nm},205\,\text{nm}]$, which corresponds to a sharp Fabry-Perot
resonance leading to low transmission values (see Fig. 3 (a)). With this
library of nanopillars, we fabricated a quadratic metalens with a diameter
$D=500\,\mu$m and a discretization parameter $\xi=0.81$, corresponding to a
smallest super-cells containing $N=2$ nanopillars and leading to a focal
length $f=203\,\mu$m (see Eq. (13)). The manufacturing of the metalens is done
using electron beam lithography, and deep reactive ion etching, using a
pseudo-Bosch process 44, providing smooth sidewalls with minimized tappering
(more details in the Methods section). In Fig. 3 (b-d), we show representative
scanning electron microscope (SEM) images of different parts of the metalens
taken with different magnifications.

#### Optical characterization

We explore the performance of our fabricated quadratic metalens by a
comprehensive analysis of the experimental point spread function (PSF), an
observable that fully describes the imaging properties of an optical system.
The optical setup used for this analysis is depicted in Fig. 4 (a) (more
details in the Methods section). We first show the normalized PSF images
produced by a laser beam illumination for $0^{\circ}{}$, $30^{\circ}{}$ and
$50^{\circ}{}$ AOI, in Figs. 4 (b), (c) and (d), respectively. One can see
good quality focal spots even for large angles, though widening of the PSF
across the horizontal direction ($x$-axis) is apparent at $50^{\circ}{}$ AOI.
We also show the light intensity distribution across the OA for $0^{\circ}{}$,
$30^{\circ}{}$ and $50^{\circ}{}$ in Fig. 4 (e), (f) and (g), respectively.
This is done by scanning the metalens position along the $z$-axis and
recording multiple PSF that are then stitched to reconstruct the corresponding
$x$-$z$ intensity maps for different AOI. One can see that the position of the
maximum intensity along the optical axis remains largely unaltered, even for
larger angles.

Figure 4: Metalens optical characterization. (a) Schematic of the optical
setup for characterization. (b), (c) and (d) Normalized PSF images for
$0^{\circ}{}$, $30^{\circ}{}$ and $50^{\circ}{}$ AOI. The scale bars
correspond to $1\,\mu$m. (e), (f) and (g) Normalized intensity distributions
measured in the $x$-$z$ plane for $0^{\circ}{}$, $30^{\circ}{}$ and
$50^{\circ}{}$ AOI. (h) Lateral displacement of the PSF in the $x$-direction
(green dots) as a function of the AOI (green dots). The theoretical prediction
[see Eq. (7)] is also shown (black dashed line). (i) Focusing efficiency (red
dots, left axis) and FWHM of the PSF (blue dots, right axis) as a function of
the AOI, together with the corresponding numerical simulations (open red and
blue circles, respectively). The lines are guide-to-the-eye. The FWHM
corresponding to a diffraction-limited lens with NA$\simeq 0.71$ [see Eq. (6)]
is also shown for reference (blue dashed line). The critical angle
$\theta^{\text{cut}}_{\text{i}}$ [see Eqs. (15) and (16)] is indicated by the
vertical black dashed line.

To provide a more quantitative picture of the metalens performance, we
measured the PSF lateral displacement for an extended range of AOI up to
$80^{\circ}{}$, as shown in Fig. 4 (h). One can see a perfect agreement
between the experimental measurements (green dots) and the analytical formula
given by Eq. (7) (black dashed line). We also measured the focusing efficiency
(red dots), and the FWHM of the PSF (blue dots) as a function of the AOI, as
shown in Fig. 4 (i). As introduced in Section Focusing properties, the former
is defined as the power within the focal spot (integrated over the second Airy
disk) divided by the power passing through the effective working area of the
lens, with diameter $D^{\text{eff}}=2f$. In the plot, one can see that the
efficiency remains virtually unchanged up to $30^{\circ}{}$ and slowly
decreases for higher angles, due to the fact that the effective working area
is getting increasingly cropped by the physical size of the lens as the AOI
increases. We compare these results with Fourier optics simulations of a
quadratic metalens with the same lens size $D=500\,$nm and focal length
$f=203\,\mu$m, and discretized transmissions and phases with pitch
$p=300\,$nm, whose values are taken from Fig. 3 (a). While the general trend
in the experiment is similar to the numerical simulations (open red circles),
the absolute values of the efficiency are smaller than the theoretical ones,
which we attribute to the fabrication imperfections. As far as the FWHM is
concerned, we measured a FWHM of about $1\,\mu$m at normal incidence,
perfectly correlating with the numerical simulations (open blue circles). With
increasing AOI, however, one can see a slight increase of the FWHM, both
experimentally and theoretically, but less pronounced in theory. The increase
of the FWHM of the PSF can be explained by the fact that coma aberrations are
introduced when the effective working area is getting cropped, thus affecting
the quality of the focal spot, in accordance with our aberration analysis of
quadratic metalenses (see SI Section 1). The discrepancy between the
experiment and theory might come from the AOI dependence of the nanopillar
transmissions and phases (see SI Section 5), as well as the vectorial nature
of light, that are not taken into account in these numerical simulations.

#### A note on the criterion for field-of-view

The actual FOV of our quadratic metalens exceeds by far the criterion given by
Eq. (16). According to this criterion, with the current discretization
parameter $\xi=0.81$, the minimum FOV after which the effective working area
is getting cut by the physical size of the lens is $\text{FOV}\geq
27^{\circ}{}$ (denoted by the vertical black dashed line in Fig. 4 (i)). As
can be seen, however, the experiment indicates that both the efficiency and
the quality of the focus are still reasonable even after the minimum AOI has
doubled. This is actually not surprising since, as explained in Section
Focusing properties, the quadratic metalens produces a PSF that is not
diffraction-limited, due to its intrinsic spherical aberrations. For our
design, the PSF has a FWHM of $1\,\mu$m, at least at normal incidence, which
corresponds to the PSF produced by a diffraction-limited system with an
$\text{NA}\simeq 0.37$, which is much smaller than the effective NA of
quadratic metalenses: $\text{NA}^{\text{eff}}\simeq 0.71$ (see Eq. (6)). In
other words, the area that contributes to focusing is much smaller than the
effective working area. Also, this is why the focal spot is not affected even
when the effective working area starts to be cropped by the metalens, which
was the criterion used to derive Eq. (16). Therefore, as we can see, quadratic
metalenses might have a FOV exceeding this criterion, provided that their
physical diameter $D$ is larger than the effective diameter
$D^{\text{eff}}=2f$.

## Imaging with a quadratic metalens

In this Section, we use the fabricated quadratic metalens in different imaging
configurations, unveiling the fundamental limitation that constrains their
performance, despite the extremely large FOV: the barrel distortion or fish-
eye effect. We first discuss this effect on the imaging of a ruler, and
illustrate the existing trade-off between the maximum FOV of an object that
can be imaged and the optical spatial resolution of the metalens. With these
limitations in mind, we then apply our metalens to the problem of fingerprint
capturing, which requires quite an extreme imaging configuration in terms of
distance and FOV.

### Barrel distortion

We start by imaging a ruler located at a distance $d$ from the metalens that
verifies $d\gg f$, in order to be in the conditions of imaging a planar object
into a planar image (see Table 1). The situation is illustrated in Fig. 5 (a)
(not to scale). We consider two configurations: $d=3\,$cm and $d=2\,$cm. The
resulting images obtained with our fabricated metalens are shown in Figs. 5
(b) and (c), respectively. One can see that parts of the ruler of sizes
approximately equal to $h\sim 12\,$cm and $h\sim 8\,$cm are being imaged,
respectively. This corresponds to an actual $\text{FOV}>120^{\circ}{}$ in both
configurations, which is estimated using the following formula (valid when
$d\gg f$):

$\text{FOV}\simeq 2\,\text{arctan}\left(\frac{h}{2d}\right)\;.$ (17)

Figure 5: Illustration of the barrel distortion effect on the imaging of a
ruler. (a) Parameters of the imaging configuration: $f$ is the focal length of
the metalens; $d$ is the distance ruler-metalens; $x_{\text{o}}$ denotes the
distance from the OA of a given object point to be imaged; $x_{\text{i}}$ is
the distance from the OA of the corresponding image point; $h$ is the size of
the ruler’s part being imaged. (b), (c) Images of a ruler taken with our
quadratic metalens (focal lens $f=203\,\mu$m) in the following configurations:
$d=3\,$cm and $d=2\,$cm, respectively. The white arrows correspond to parts of
the ruler that are being imaged but not entirely resolved, and the red arrows
correspond to the parts of the ruler for which the ruler graduations are fully
resolved, and their lengths $h$ have been calculated from Eqs. (18) and (17)
by taking the object feature size $\Delta x_{\text{o}}\sim 1\,$mm and the
image spatial resolution $\Delta x_{\text{i}}\sim 2\,\mu$m.

In Figs. 5 (b) and (c), the barrel distortion is visible, which is
characteristic of wide-FOV lenses (_e.g._ fisheye lenses) 45, 46, 47. This
follows from the fact that equally spaced object points produce image points
that get more compressed in the edges of the image. The barrel distortion in
itself is not a problem for imaging since it can be compensated with a post-
processing treatment (see next Section). However, as the distance of the
object points $x_{\text{o}}$ increases, or in other words, as the FOV of the
object increases (according to Eq. (17) with $h=2x_{\text{o}}$), the
corresponding image points become increasingly compressed (see SI Section 6
and Fig. S7). Subsequently, the image spatial resolution required to resolve
these points will increase. In practice, the image spatial resolution
corresponds, depending on which one is the limiting factor, to either the
detector spatial resolution (related to the pixel pitch), or to the optical
spatial resolution of the lens (proportional to the FWHM of the PSF), which is
its ability to resolve details. In order to be more quantitative, we derived
explicitly (see SI Section 6 and Fig. S8) the maximum FOV of an object over
which features with a given minimum size $\Delta x_{\text{o}}$ can be resolved
with a quadratic metalens, and for a given image spatial resolution $\Delta
x_{\text{i}}$:

$\text{FOV}=2\,\text{arctan}\left(\sqrt{\left(\frac{f}{d}\frac{\Delta
x_{\text{o}}}{\Delta x_{\text{i}}}\right)^{\frac{2}{3}}-1}\right)\;.$ (18)

In view of this, we analyse the previous ruler imaging experiment. We take the
minimum object feature size to be $\Delta x_{\text{o}}\sim 1\,$mm. Since in
our imaging configuration, we are not limited by the detector spatial
resolution (for the details of the imaging experiment, see SI Section 7 and
Fig. S9), we take for the image spatial resolution the worse FWHM of the PSF,
that we measured to be $\Delta x_{\text{i}}\equiv\text{FWHM}\sim 2\,\mu$m
(obtained for an AOI of $80^{\circ}{}$, as seen in Fig. 4 (b)). The
application of Eq. (18) gives the maximum FOV of the ruler over which such
$1\,$mm features can be resolved. They turn out to be $\text{FOV}\simeq
96^{\circ}$ in the configuration with $d=3\,$cm and $\text{FOV}\simeq
109^{\circ}$ in the configuration with $d=2\,$cm, corresponding to portions of
the ruler with sizes equal to $h=6.7\,$cm and $h=5.6\,$cm, respectively
(according to Eq. (17)). One can indeed check with naked eye on the images of
Figs. 5 (b) and (c) that the ruler graduations can only be properly resolved
for those parts of the rulers with sizes $h<7\,$cm and $h<6\,$cm,
respectively.

This example illustrates that it is the barrel distortion that limits the FOV
of the lens in an imaging configuration, which, in turn, is ultimately
constrained by the image spatial resolution required to resolve details of an
object. In the example of the ruler located at two different distances
$d=3\,$cm and $d=2\,$cm from our fabricated lens, while portions of the ruler
with sizes $h\sim 12\,$cm and $h\sim 8\,$cm, respectively, can be imaged
thanks to the large FOV of the metalens, its optical spatial resolution only
allows to resolve parts with sizes $h=6.7\,$cm and $h=5.6\,$cm, respectively,
with the desire $1\,$mm spatial resolution.

### A practical case: fingerprint imaging

In this section, we investigate more extreme imaging configurations (i.e.
shorter distances $d$) in the particular problem of fingerprint capturing, to
demonstrate the imaging potential of quadratic metalenses. Indeed, the
subwavelength scale of the metalens opens up new possibilities to realize
compact optical detection systems possessing large FOV (see the artistic
picture on Fig. 6 (a), left panel), which remains elusive for conventional
refractive optics, as it typically involves complex optical trains. In order
to scale down the device thickness, the object-lens distance $d$ should be
reduced as much as possible, which in turn requires higher FOV for a given
fingerprint size. Taking into account the barrel distortion constraint, the
object-lens distance obviously possesses a lower limit, below which one can no
longer resolve the object features. Here, we examine this limit with our
fabricated metalens by imaging a $5\,\text{mm}\times 5\,$mm fingerprint
featuring details $\sim 100\,\mu$m.

Figure 6: Fingerprint imaging. (a) Artistic picture of the fingerprint
capturing device and original image of the printed fingerprint (not to scale).
(b), (c) Experimental images of the fingerprint taken with our fabricated
metalens at distances $d=2.5\,$mm and $d=1.5\,$mm, respectively. (d), (e)
Corresponding simulated images. (f), (g) Post-processed experimental images
(b) and (c), respectively. The scale bars correspond to $100\,\mu$m.

The fingerprint (see Fig. 6 (a)) was printed on a white paper and illuminated
by a laser in a transmission configuration. Because of the small focal lens of
our metalens and the bulky detector that we used, we had to transfer the image
produced by the metalens to a CCD camera using a free-space microscope setup
with 5X magnification (for the details of the imaging experiment, see Methods
and SI Section 7 and Fig. S9). In applications, however, in order to achieve a
compact system, it is not desirable to use a microscope and one would place a
small detector directly in the image focal plane of the metalens instead.
Figs. 6 (b) and (c) show the experimental images for two configurations:
$d=2.5\,$mm and $d=1.5\,$mm, respectively. We also provide the corresponding
simulated images in Figs. 6 (d) and (e). The imaging simulation was done using
Fourier optics and assuming a perfect quadratic phase profile for the metalens
(see Methods). Moreover, in order to mimic the incoherent imaging, the
fingerprint amplitude mask was multiplied with a random phase mask and
averaged over realizations. One can see a good correspondence between the
simulated images and the experimental ones. As expected, all the images
clearly present the barrel distortion effects. However, by a proper post-
processing (see Methods), it is possible to retrieve corrected images from the
experimental ones, as shown in Figs. 6 (f) and (g). One can see that the whole
$5\,\text{mm}\times 5\,$mm fingerprint area becomes visible at $d=2.5\,$mm,
while for the case of $d=1.5\,$mm, the edges are blurred by the barrel
distortion and the corresponding information is lost, and thus only a
fingerprint area of about $3\,\text{mm}\times 3\,$mm is detected. In the
former case, taking into account the distance between the lens and the image
plane, the total thickness of the device is still only $\sim 2.7\,$mm, making
for a very compact system for fingerprint detection.

In principle, of course, the object-lens distance $d$ could be further
decreased in an attempt to make an even more compact device. This, however,
comes at the cost of imaging a smaller object size. To do so, the focal length
should be decreased accordingly, in order to remain in the regime $d\gg f$ of
imaging a planar object into a planar image. To illustrate this scenario, we
fabricated another metalens with smaller focal length while keeping the same
diameter/focal length ratio ($f=100\,\mu$m, $D=246\,\mu$m). Using this lens,
we imaged the fingerprint at the smallest possible distance $d=800\,\mu$m,
which corresponds to the thickness of the substrate (see Methods, SI Section 7
and Fig. S9).

## Conclusion

In this work, we have shown that in terms of imaging, a quadratic metalens
images a _planar_ object along an _ellipsoid_ surface in general, and that in
order to produce a _planar_ image from a _planar_ object, the object must be
located at a distance $d$ from the metalens that satisfies $d\gg f$, $f$ being
the focal length of the metalens. In this case, the image has a maximum cross-
section of $2f\times 2f$, which also sets the maximum size of the sensor (if
no magnification system is used). Therefore, one must keep in mind that while
quadratic metalenses can favor miniaturization (and also cost matters since
smaller sensor sizes are cheaper), they can also decrease the resolution, so
that only for applications which do not require large sampling, like _e.g._ in
the imaging of fingerprints, one can benefit from them. On the other hand, the
relatively large depth-of-focus of quadratic metalenses could be beneficial
for certain applications, such as in biological imaging, where the depth-of-
focus matters to obtain morphological information about tissues for example
48. We should also point out here that the image contrast can be improved by
placing an aperture stop before the quadratic metalens (see _e.g._ , Ref. 49,
and a discussion of such configuration based on a modulation transfer function
analysis in SI Section 8 and Fig. S10).

We also confirmed that quadratic metalenses are excellent flat optical
solutions for wide FOV imaging. Indeed, we designed and fabricated a sample
working in the near IR, which, despite a rather low focusing efficiency (about
$14\%$ in theory and $8\%$ experimentally) which is peculiar to this type of
phase profile, allowed us to image a ruler over a FOV of about
$100^{\circ}{}$. In addition, we highlighted that it is the optical resolution
of the metalens and/or the detector resolution that prevents the FOV to be
larger, which stems from the intrinsic barrel distortion, or fisheye effect.

As an example and proof-of-concept of an imaging configuration in which the
benefits of these metalenses are exploited, we show, using our fabricated
sample, the imaging of a fingerprint of size $\sim 5\,$mm for a distance
object-lens as small as $d=2.5\,$mm, with which we are able to resolve
features with sizes of the order of $100\,\mu$m, resulting in a total device
thickness of $\sim 2.7\,$mm. This demonstrates the ability of quadratic
metalenses for imaging in extreme configurations of large objects at
relatively short distances.

## Methods

##### Fourier optics calculations:

The Fourier optics calculations used in this work are based on the _plane-wave
spectrum method_ (see _e.g._ 50, Chapter 3). The so-called plane-wave spectrum
or angular spectrum representation is obtained by applying scalar wave
approximation and solving the Helmholtz equation. The method is implemented
using transfer function representation in order to exploit discrete Fourier
transforms and multiplicative relationships in frequency domain for the
calculation of field propagation. Parabolic lens profiles, defined by
wavelength of operation, focal length and lens diameter according to Eq. (3),
are numerically characterized using this plane wave decomposition technique.
In oblique angle of incidence calculations, complex fields of an incident
plane wave are superimposed with fields corresponding to the parabolic phase
profile. Subsequent propagation of the fields comprises Fourier transform,
multiplication by transfer function and inverse Fourier transform.
Transmissive areas of the parabolic lens are visualized by applying a low-pass
filter to reduce spatial frequencies to propagating components. Calculations
illustrating light propagation in the $z$-direction are reduced to two
dimensions ($x$ and $z$) in order to reduce computational cost. Point-like
sources with a high angular spread are realized using a Gaussian intensity
profile with a sub-wavelength FWHM of $500\,$nm. Angle dependent efficiency of
the lens as well as FWHM and spatial position of foci are calculated in the
focal plane given by $z=194\,\mu$m. Efficiency is calculated as the ratio
between the power incident on the total lens area and the power in a circle up
to the second Airy minima. For the imaging configurations, a single snapshot
of incoherent light propagation is modeled by assigning a pseudo-random phase
to each pixel in the object plane and recording propagated fields in the image
plane. The full image is reconstructed in a Monte Carlo like simulation in
which multiple snapshots using different random phase masks are averaged. It
is important to note that these Fourier optics simulations require a fine
sampling of the object and image planes. In all calculations, we used the
design wavelength of $740\,$nm, a spatial resolution of at least $150\,$nm and
lateral calculation domain size of at least $1\,$mm $\times$ $1\,$mm.

##### FDTD simulations:

We use Lumerical FDTD software for the FDTD simulations of the transmission
efficiencies and phase-delays of periodic arrays of silicon nanopillars with
identical diameters. We consider an hexagonal lattice with lattice constant
$p=300\,$nm (center-to-center distance between closest nanopillars), and we
use periodic boundary conditions at normal incidence or Bloch boundary
conditions at oblique incidences. The pillars are lying on a glass substrate
of refractive index $n=1.46$. We simulate a monochromatic plane wave with
wavelength $\lambda_{0}=740\,$nm and amplitude equal to unity, coming from the
glass substrate (incident medium $n_{\text{i}}=1.46$), and record the
transmission efficiencies and phases after the propagation through the
nanopillars in air (transmission medium $n_{\text{t}}=1$), and we sweep over
the nanopillar diameters and/or AOI.

##### Fabrication:

To fabricate our samples, we first deposit a $350\,$nm thick layer of
amorphous silicon on $800\,\mu$m thick fused silica substrates using chemical
vapor deposition technique (model Producer SE, Applied Materials) followed by
a $30\,$nm thick layer of chromium (Cr) by evaporation (Evovac, Angstrom
Engineering) to act later as a hard mask for silicon etching. For the mask
nanopatterning, a thin layer of hydrogen silsesquioxane (HSQ, Dow Corning,
XR-1541-002) is formed by spin-coating and baking. Electron-beam lithography
(Elionix, 100 kV) is used to expose the negative resist, and the development
in Tetramethylammonium hydroxide solution (TMAH, 25%) reveals the metalens
design. The pattern is then transferred from the resist to the Cr hard mask
using ICP-RIE (Plasmalab System 100, Oxford Instruments) with chlorine
chemistry. The choice of using Cr as mask for silicon etching is to get rid of
the remaining HSQ. The resist thickness ($50\,$nm) is then chosen to no longer
survive the silicon etching. To obtain anisotropic etching with smooth
sidewalls, the samples are etched using the so-called pseudo-Bosch process, in
which etching (SF6 gas) and passivation (C4F8 gas) chemistries are used
simutaneously. Optimization of the chamber (SPTS Technologies) parameters
allows precise control of the sidewall angle of the nanopillars. The chromium
mask is then removed by Cr etchant (Sigma-Aldrich).

##### Optical measurements:

We performed the PSF optical characterization using a tunable fiber laser
centered at 740 nm wavelength (SuperK NKT Photonics equipped with a tunable
single line filter SuperK Select). The fiber output together with collimation
optics were placed on a rotation stage (see Fig. 4 (a)). A linear polarizer
was utilized to produce a TM-polarized laser beam. The PSF of the metalens was
imaged by a free-space 83$\times$ microscope setup (100$\times$ Olympus plan
apo infinity-corrected objective, NA=0.95, a tube lens with 150 mm focal
length and CS895MU Thorlabs CMOS camera). The focusing efficiency was
calculated by integration of the PSF intensity over a circular aperture with a
constant radius, which corresponds to the second Airy minimum position for AOI
= $0^{\circ}$. Anti-reflection coated achromatic doublet (Thorlabs
AC254-030-B-ML) was utilized as a reference lens.

For imaging the ruler and fingerprint, we used a free-space 5$\times$
microscope setup depicted on SI Fig. S9. The objects were illuminated by the
laser in transmission configuration. To produce a line pattern for the ruler
illumination, we used a Thorlabs $0.4^{\circ}\times 100^{\circ}$ line
engineered diffuser. The post-processing of the fingerprint images was
performed by the following algorithm:
1\. The radius $R$ and the center of distortion were extracted from the raw
image. The maximum radius of new undistorted image was calculated as
$R_{\text{max}}=f\,\text{tan}(\text{arcsin}(R/f))\;;$
2\. The undistorted image pixels $(i,j)$ were filled in by the intensity
values extracted from the raw image $(m,k)$. $(m,k)$ pixels were identified by
the following formula:
$m,k=\frac{i-R_{\text{max}}}{\sqrt{(i-R_{\text{max}})^{2}+(j-R_{\text{max}})^{2}}}\,f\,\text{sin}\left\\{\text{arctan}\left(\frac{\sqrt{(i-R_{\text{max}})^{2}+(j-R_{\text{max}})^{2}}}{f}\right)\right\\}+R-1\;;$
3\. The new undistorted image was binarized using an adaptive thresholding
technique suitable for a non-uniform intensity distribution.

## Authors’ contributions:

E.L. designed the metalens, made the FDTD simulations, and carried out the
theoretical analysis of the imaging properties. T.W.W.M. implemented the
Fourier optics code, made all the Fourier optics calculations (unless
otherwise stated), and took part in the theoretical analysis of the imaging
properties. D.E. optimized the fabrication process and fabricated all the
samples. A.V.B. performed the optical characterization, the imaging
experiments, the Fourier optics calculations for the fingerprint and the
calculations of the modulation transfer function. E.K. performed the
aberration analysis of the quadratic metalenses based on Zernike polynomials.
S.L. supervised the fabrication. R.P.-D. implemented the ray-tracing code.
R.P.-D. and A.I.K. conceived the idea and supervised the work. E.L., T.W.W.M.,
D.E. and A.V.B. contributed equally to this work and wrote the manuscript. All
the authors read and reviewed the manuscript.

The authors thank Steven Lee Hou Jang for depositing the amorphous silicon on
a $12"$ fused silica wafer, and Moitra Parikshit for providing accurate values
of the refractive index and extinction coefficient of amorphous silicon via
ellipsometry measurements. The authors acknowledge financial support from
National Research Foundation of Singapore (grant NRF-NRFI2017-01), IET A F
Harvey Engineering Research Prize 2016, A*STAR SERC Pharos program (grant 152
73 00025) (Singapore), and AME Programmatic Grant A18A7b0058 (Singapore).

Analysis of the monochromatic aberrations of quadratic metalenses: Zernike
decomposition of the wavefront and ray-tracing pictures for different AOI.
Fourier optics calculations of the focal spot position, focusing efficiency
and FWHM as a function of the AOI. Imaging theoretical analysis: derivation of
the ellipse and hyperbola equations in the paraxial approximation. Plot of the
minimum metalens FOV as a function of the discretization parameter/unit-cell
pitch. Additional information on the transmission and phase of the meta-atoms
for different AOI and for TM/TE polarizations: transmission and phase maps for
periodic array of unit-cells, diffraction efficiencies into the blazed order
for periodic array of super-cells. Derivation and plot of the equation for the
maximum object FOV in terms of the image spatial resolution. Additional
information about the imaging experiment of the ruler and the fingerprint.
Analysis of the monochromatic modulation transfer function.

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