[1]Shaban B. Sulejman

[1]Ann Roberts

1]ARC Centre of Excellence for Transformative Meta-Optical Systems, School of
Physics, The University of Melbourne, Victoria 3010, Australia

2]Dorevitch Pathology, Heidelberg, Victoria 3084, Australia

3]ARC Centre of Excellence in Exciton Science, School of Chemistry, The
University of Melbourne, Victoria 3010, Australia

4]Florey Institute of Neuroscience and The University of Melbourne, Parkville,
VIC 3010, Australia

5]St Vincent’s Private Hospital, East Melbourne, Victoria 3065, Australia

6]School of Science, RMIT University, Melbourne, Victoria 3001, Australia

7]ARC Centre of Excellence for Transformative Meta-Optical Systems (TMOS),
Department of Electronic Materials Engineering, Research School of Physics,
The Australian National University, Canberra, ACT 2601, Australia

8]ARC Centre of Excellence for Transformative Meta-Optical Systems, Department
of Electrical and Electronic Engineering, The University of Melbourne,
Victoria, 3010, Australia

# Metasurfaces for infrared multi-modal microscopy: phase contrast and bright
field

<EMAIL_ADDRESS>Lukas Wesemann  Mikkaela McCormack  Jiajun Meng
James A. Hutchison  Niken Priscilla  Gawain McColl  Katrina Read  Wilson Sim
Andrey A. Sukhorukov  Kenneth B. Crozier<EMAIL_ADDRESS>[  [  [
[  [  [  [  [

###### Abstract

Different imaging modalities are used to extract the diverse information
carried in an optical field. Two prominent modalities include bright field and
phase contrast microscopy that can visualize the amplitude and phase features
of a sample, respectively. However, capturing both of these images on the same
camera typically requires interchanging optical components. Metasurfaces are
ultra-thin nanostructures that can merge both of these operations into a
single miniaturized device. Here, a silicon-based metasurface that supports a
Mie resonance is demonstrated to perform near-infrared phase contrast and
bright field multi-modal microscopy that can be tuned by changing the
polarization of the illumination. We performed experiments using optical
fields with phase variations synthesized by a spatial light modulator and
introduced by propagation through semi-transparent samples, including C.
elegans, unstained human prostate cancer cells and breast tissue. The results
demonstrate the potential of metasurfaces for label-free point-of-care
testing.

###### keywords:

Metasurfaces, microscopy, phase contrast imaging, biological imaging, Mie
resonances, image processing

## Introduction

Multi-modal microscopy is a powerful tool for acquiring images with a diverse
range of imaging techniques using a single multi-functional system [1, 2].
This includes, but is not limited to, confocal microscopy [3], super-
resolution imaging [4, 5], tomography [6] and fluorescence microscopy [7]. Its
versatility has been evident across a wide range of applications, from
analyzing meteorite structures [8] to advanced biomedical imaging [9, 10].
Bright field microscopy stands as the simplest and most widely used imaging
technique for examining microscopic samples that introduce amplitude
variations through absorption. Despite its popularity, it cannot capture the
phase variations of light essential for visualizing transparent samples, such
as unstained cells. Phase microscopy techniques, including Zernike [11] and
differential interference contrast (DIC) microscopy [12], can detect features
attributed to varying refractive index or thickness in transparent samples and
quantitative methods can measure these phase variations [13]. While various
methods of phase microscopy have been integrated with other imaging modalities
[14, 15, 16, 17, 18], its integration with bright field microscopy provides a
comprehensive sample visualization [19, 20, 21, 22, 23] and information that
can be key to quantitative phase imaging strategies [24]. However, the
complexity, size and cost of the associated equipment have largely limited
their availability to specialized laboratories, precluding access to point-of-
care diagnostics and in developing countries or remote locations.

The recent COVID-19 pandemic saw a rise in demand for portable and rapid
diagnostic systems [25]. There have been first experiments towards compact
imaging systems that have attracted attention for applications beyond the
laboratory [26]. Metasurfaces, defined as thin photonic devices with sub-
wavelength structure over a 2D surface, have emerged as promising solutions
[27]. Flat meta-lenses are being integrated into portable devices [28] and
other metasurfaces are finding use in areas including phase imaging [29, 30]
and multi-modal microscopy [31, 32, 33]. For example, a metasurface was used
to synchronously generate spiral phase contrast and bright field images [34],
albeit requiring two redirected beams with additional optical components.
Recent developments have also centered on adding tunability to these devices
[35], leveraging techniques such as phase-change materials [36, 37, 38, 39,
40], liquid crystals [41], opto-fluidics [42], mechanical straining [43] and
electrical gating [44]. As an external parameter within an imaging system,
polarization also presents an attractive approach to optical tunability [45,
46]. It has proved instrumental for dynamically controlling color filtering
[47], lensing [48] and phase imaging [49, 50].

Here, we present the first demonstration of a metasurface designed for
polarization-tunable, near-infrared phase contrast and bright field multi-
modal microscopy. The metasurface consists of rectangular blocks of silicon on
a glass substrate. When an incident electric field has a component parallel to
the long axis of the blocks, the metasurface generates phase contrast images
via a Mie resonance. However, bright field images are produced under the
orthogonal polarization, enabling bi-modal switching with a switching contrast
of up to 93%. Importantly, this does not require interchanging components into
and out of the system nor the use of beam-splitters, as it is usually the case
in multi-modal microscopy methods. We experimentally demonstrate this with
Caenorhabditis elegans samples, as well as unstained human prostate cancer and
breast tissue samples. Pathologists commonly use chemical stains, such as
haematoxylin and eosin (H&E) [51], to image these samples because imaging
breast tissue, for example, can be difficult due to small ($\Delta n\sim
10^{-2})$ refractive index difference relative to the surrounding [52].
However, staining is an irreversible process that introduces additional costs
and can delay the turnover time in pathology. We show that our proposed
imaging method can be used to distinguish between breast ducts and blood
vessels in unstained breast tissue derived from a patient with breast cancer.
Moreover, the use of near-infrared illumination offers deeper penetration into
tissue compared to visible light [53], while retaining the use of low-cost
silicon photo-detectors. The dual approach of polarization switchability and
infrared illumination offers an advance in miniaturized multi-modal
microscopy. It holds the potential to deliver prompt results where existing
technologies are unavailable and aligns with the need for compact imaging
systems in point-of-care settings [54].

## Results

### Tunable multi-modal microscopy - principles

The fundamental concept for the application of the metasurface to tunable
multi-modal microscopy is the filtering of the spatial frequencies of an image
through object-plane image processing. The optical transfer function of the
metasurface, defined as the Fourier transform of its Green’s function,
characterizes its response to the spatial frequencies of an input image. In
general, it is a rank-2 tensor given in the $\\{p,s\\}$-polarization basis as,

$H(k_{x},k_{y})=\begin{pmatrix}H_{pp}(k_{x},k_{y})&H_{ps}(k_{x},k_{y})\\\
H_{sp}(k_{x},k_{y})&H_{ss}(k_{x},k_{y})\\\ \end{pmatrix}\ .$ (1)

Henceforth, we utilize an $\\{x,y\\}$ basis, i.e. $H_{xx}$, $H_{xy}$, $H_{yx}$
and $H_{yy}$, describing the impact of the metasurface on the transverse
components of the incident field. By taking the $z$-axis as the optical axis,
$k_{x}$ and $k_{y}$ denote the transverse spatial frequency components of the
wave-vector $\vec{k}$. Ignoring vector effects for now for illustration, the
impact of the metasurface on the spatial frequencies of an input image of a
semi-transparent sample $O(x,y)=O_{0}(x,y)e^{i\varphi(x,y)}$, where $O_{0}$ is
the amplitude and $\varphi(x,y)$ is the phase, can be modelled using the
convolution theorem,

$I(x,y)\approx\left|\mathscr{F}^{-1}\left\\{H(k_{x},k_{y})\tilde{O}(k_{x},k_{y})\right\\}\right|^{2}\
.$ (2)

Here, $I$ is the output intensity image, $\mathscr{F}$ denotes the Fourier
transform and $\tilde{O}=\mathscr{F}\\{O\\}$. Note that $|O(x,y)|^{2}$ is the
bright field (absorption) image that is independent of $\varphi(x,y)$. If we
let the $x$-axis be aligned with the long axis of the silicon blocks for an
un-tilted metasurface (normal incidence), then the metasurface has an
approximately linear optical transfer function at a tilt of
$2\text{\,}\mathrm{\SIUnitSymbolDegree}$ about the $y$\- (short) axis (Fig.
1c) for $x$-polarized illumination, i.e. $H_{xx}\propto k_{x}+k_{x0}$, where
$k_{x0}$ is a constant. This can compute the derivative along the
$x$-direction, plus a constant offset that retains a non-zero background in
the output image. In the case of a pure phase sample $|O(x,y)|\propto
O_{0}=\text{constant}$, the sample is visualized because the phase derivatives
are converted into intensity variations, e.g.

$I(x,y)\approx\left|O_{0}\left(\frac{\partial\varphi(x,y)}{\partial
x}+k_{x0}\right)\right|^{2}\ .$ (3)

One could also uniquely distinguish the phase derivatives as they will
manifest as different intensity levels that are above or below the background
(region of zero phase gradient), i.e. $I(x,y)\approx|O_{0}k_{x0}|^{2}$. On the
other hand, the metasurface is approximately transparent when the illumination
is $y$-polarized, which corresponds to a flat optical transfer function,
$H_{yy}\approx\text{constant}$. In this case, the output image is proportional
to the bright field image, $I(x,y)\propto|O(x,y)|^{2}$. Therefore, with this
configuration, $x$-polarized illumination leads to the production of phase
contrast images, while $y$-polarized illumination leads to bright field
images. It is shown in the Supplementary Information S2.1 that the effects of
cross-polarization are negligible, i.e. $H_{xy}$ and $H_{yx}$ in this case are
zero, and we ignore diffraction effects.

We define the switching contrast of the metasurface as
$\delta=H_{yy}(0,0)-H_{xx}(0,0)$ and the numerical aperture (NA) of the
metasurface as the width of the optical transfer function in $k_{x}$ where the
above conditions are met. This is in contrast to the numerical aperture of the
imaging system in which the metasurface is used. Other performance metrics of
the metasurface are given in the Supplementary Information S2.2, as well as
further details of the theoretical principles (S1.1) and polarization used in
this work (S3.1).

### Design and optical response

Figure 1: Design of the metasurface. (a) A stylized depiction of use of the
metasurface to perform tunable multi-modal microscopy, with experimental
images of a mock red blood cell (erythrocyte) given in the insets. (b) The
composition of the metasurface shown in 3D (metasurface tilted) (i) and bird’s
eye (metasurface un-tilted) (ii) views of the unit cell. A schematic of the
polarization is also given in i. (c) The fabricated coverslip with patterned
regions of the metasurface. (d) Scanning electron microscope images of the
metasurface.

A schematic of the approach used here is given in Fig. 1a. The metasurface
consists of a 2D lattice of rectangular silicon resonators of length
$L=$322\text{\,}\mathrm{nm}$$, width $W=$293\text{\,}\mathrm{nm}$$ and height
$H=$236\text{\,}\mathrm{nm}$$ (Fig. 1b). The unit cell of the structure is
repeated with a lattice constant of $P=$526\text{\,}\mathrm{nm}$$ in both
directions (Fig. 1b-ii). The fabricated metasurface was patterned on a glass
wafer $15\text{\,}\mathrm{mm}$ by $15\text{\,}\mathrm{mm}$ in size (Fig. 1c)
and the total size of each metasurface was
$760\text{\,}\mathrm{\SIUnitSymbolMicro m}$ by
$507\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The fabrication steps are
outlined in the Methods section. Scanning electron microscope images of the
fabricated metasurface reveal the rectangular geometry of the resonators (Fig.
1d and Supplementary Information S4.1), which is the basis of the polarization
tunability.

Figure 2: Optical response of the metasurface. ((a)-i) The simulated normal-
incidence transmission spectrum of the metasurface for $x$\- and
$y$-polarization. ((a)-ii) Same as a-i but for experiment. ((a)-iii) The
simulated multipole decomposition of the normal-incidence transmission
spectrum under $x$-polarization. The electric dipole resonance is indicated by
the grey shaded region. ((b)-i) The magnitude of the simulated co-polarized
components of the optical transfer function along $k_{y}=0$ at the operating
wavelength for $x$\- and $y$-polarization. ((b)-ii) Same as b-i but for
experiment. ((b)-iii) Linear fitting of the optical transfer function in b-i.
(c) The simulated magnitude (heat map) and direction (arrows) of the electric
field (top), and the electromagnetic energy density (heat map) and time-
averaged Poynting vector (arrows) (bottom) in the $xz$-plane of the unit cell.
(d) The experimental normal-incidence transmittance through the metasurface as
the polarization was switched between $x$ and $y$.

The optical response of the metasurface was investigated by measuring its
angle-dependent transmission. Simulations were carried out using a commercial
electromagnetic solver (COMSOL Multiphysics 6.1) implementing the finite
element method, which is detailed in the Methods section. The experiments were
performed using the normal-incidence and angle-dependent spectroscopy
configurations described in the Methods section and in the Supplementary
Information S3.2. The simulated (Fig. 2a-i) and experimental (Fig. 2a-ii)
normal-incidence transmission spectra of the metasurface show a minimum in the
transmission at a wavelength of $969\text{\,}\mathrm{nm}$ under $x$-polarized
illumination. This set the operating wavelength to be
$\lambda=$969\text{\,}\mathrm{nm}$$. The transmission minimum reached a level
of 0.002% in the simulation and 20% in the experiment. It also red-shifted by
$\Delta\lambda\approx$30\text{\,}\mathrm{nm}$$ in both the simulation and
experiment when the polarization was switched to $y$-polarization. This
produces a polarization-switchable transmission at the operating wavelength,
with a contrast of $\Delta=T_{y}-T_{x}=86\%$ in the simulation and
$\Delta=71\%$ in the experiment. Here, $T_{x}$ and $T_{y}$ are the
transmittance levels under $x$\- and $y$-polarized illumination, respectively.
By using the method of Ref. [55] detailed in the Supplementary Information
S1.2, a multipole decomposition of the simulated transmission spectrum under
$x$-polarization (Fig. 2a-iii) revealed that the transmission minimum was
associated with an electric dipole Mie resonance, which is discussed further
in the Supplementary Information S1.3.

Plots of the electric field (Fig. 2c-top) and power flow (Fig. 2c-bottom) at
the operating wavelength show the different responses of the metasurface.
Under $x$-polarization, the electric dipole excitation preferentially reflects
the illumination, which is apparent in the power flow and in the normal-
incidence reflectance spectrum given in the Supplementary Information S2.1. On
the other hand, the power flow is directed through the metasurface under
$y$-polarization. The quality factor of the resonance was calculated to be
$Q=17.3$ in the simulation and $Q=19.3$ in the experiments, which are of the
order of the typical values reported for Mie resonances [56]. Two tables are
given in the Supplementary Information (S2.2 and S4.3) that summarize the
various performance metrics of the metasurface. Furthermore, the
electromagnetic behavior of the metasurface at the operating wavelength in all
of the cross-sectional planes of the unit cell are also given in the
Supplementary Information S2.1.

The angle-dependent transmission through the metasurface as a function of the
polar angle of incidence $\theta$ was simulated and experimentally measured.
Here, $\theta$ represents the angle from the optical axis ($z$-axis) in the
$xz$-plane (Fig. 1c-i), corresponding to an azimuthal angle of
$\phi=$0\text{\,}\mathrm{\SIUnitSymbolDegree}$$. Using the relationship
between the angles of incidence of plane waves and the spatial frequencies of
light that is explained in the Supplementary Information S1.1, the angle-
dependent transmission amplitude through the metasurface is equivalent to the
magnitude of the relevant component of the optical transfer function [57]. The
results (Fig. 2b) show the magnitude of the co-polarized components of the
optical transfer function along $k_{y}=0$ as a function of the normalized
spatial frequency $k_{x}/k_{0}$, where $k_{0}=2\pi/\lambda$ is the wavenumber.

For $x$-polarization, the component $|H_{xx}(k_{x},0)|$ exhibits a high-pass
lineshape with a processing efficiency of
$\eta=\max(|H_{xx}(k_{x}^{\text{max}},0)|)=79\%$ in the simulation (Fig. 2b-i)
and $\eta=76\%$ in the experiment (Fig. 2b-ii). Here, $k_{x}^{\text{max}}$ is
the maximum spatial frequency component at the edge of the ‘contrast zone’,
which we define as the shaded region in Fig. 2b-i-ii where the metasurface can
be used for tunable multi-modal microscopy. It has a maximum numerical
aperture of $\text{NA}\approx 0.17$. Linear fitting of the simulated optical
transfer function within the range $k_{x}/k_{0}=0.07$ to $0.17$ is shown in
Fig. 2b-iii. The linear fit, derived using the polyfit function and curve
fitting toolbox in MATLAB, can be represented by the relation $(4.8\pm
0.11)k_{x}/k_{0}-(0.18\pm 0.014)$. On the other hand, the component
$|H_{yy}(k_{x},0)|$ was approximately flat with $k_{x}$ under $y$-polarization
in both the simulation (Fig. 2b-i) and experiment (Fig. 2b-ii). It exhibited a
switching contrast of $\delta=93\%$ in the simulation and $\delta=51\%$ in the
experiment. This difference was due to the experimental transmittance level
being 20% higher than the simulated level at normal incidence under
$x$-polarization, which corresponds to a 40% transmission amplitude
difference. Finally, the experimental switching of the metasurface at the
operating wavelength (Fig. 2d) showed consistent performance as the linear
polarization of the illumination was repeatedly switched between $x$ and $y$.
Additional simulation data is given in the Supplementary Information S2,
including the full optical transfer function tensor and transmission spectra
as functions of the polarization, the refractive index of the surrounding and
the geometric parameters of the metasurface.

### Tunable multi-modal microscopy - experiment

Figure 3: Tunable multi-modal microscopy of samples generated by a spatial
light modulator. (a) A flowchart of the images produced by the metasurface for
different polarization states of the illumination. (b) The configuration used
in the experiment. (c-top left) The amplitude mask. (c-top right) The phase
profile shown on the SLM. (c-bottom) The image taken in the absence of the
metasurface. (d) The image obtained by subtracting the image acquired under
$x$-polarization from that acquired under $y$-polarization.

Tunable multi-modal microscopy experiments were conducted on a range of
samples using the metasurface and near-infrared illumination at the operating
wavelength of $969\text{\,}\mathrm{nm}$. In all of the measurements, the
metasurface was tilted by $2\text{\,}\mathrm{\SIUnitSymbolDegree}$ to the
optical ($z$) axis, corresponding to a normalized spatial frequency of
$k_{x0}/k_{0}\approx 0.03$, about the short axis of the resonators
(illustrated in Fig. 1c and the Supplementary Information S3.1). This accessed
the contrast zone of the metasurface by introducing an offset to the optical
transfer function to become the shifted $k$-space origin about the angular
offset. Consequently, some of the background associated with low spatial
frequencies was retained in the output images, similar to that seen in DIC.
This produced the appearance of pseudo-3D contrast on a non-zero background.
All of the output images were normalized with respect to their brightest
pixels. Finally, the imaging experiments provided a field of view of
approximately $500\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and a resolution
limit of $\rho\geq
0.5\lambda/\text{NA}=$2.85\text{\,}\mathrm{\SIUnitSymbolMicro m}$$.

The first set of experiments, shown in Fig. 3b and detailed in the Methods
section, involved a computer-controlled spatial light modulator (SLM) that
replicated the phase profiles of a range of transparent samples with a phase
excursion of $\pi$ radians. This included the word ‘Phase’ (Fig. 3c-top right)
that was next to a purely absorptive amplitude mask containing the word
‘Amplitude’ (Fig. 3c-top left) placed directly in front of the SLM. The steps
taken to fabricate the amplitude mask are detailed in the Methods section. The
other phase profiles that were used on the SLM included models of a human red
blood cell (Fig. 1a) and a leukaemic Jurkat cell, which were each next to the
words ‘Erythrocyte’ and ‘T-Lymphocyte’ from the amplitude mask, respectively.
The imaging results of the cells are presented in the Supplementary
Information S2.4, as well as simulations of the imaging experiments.

The images produced by the metasurface were organized in a flowchart with
respect to the different polarization states of the illumination (Fig. 3a).
Both of the transparent (‘Phase’) and opaque (‘Amplitude’) features were
visible in the image obtained under $x$-polarization (Fig. 3a-left). The
contrast seen of the word ‘Phase’ transitioned from a maximum to a minimum as
the polarization was rotated from $x$ to $y$ (Fig. 3a-top). The opposite
effect was observed when the polarization was rotated from $y$ to $x$ (Fig.
3a-bottom). It was also observed that the contrast of the word ‘Phase’ flipped
between the two cases. This was a consequence of accessing either the positive
or negative slope of the optical transfer function. Meanwhile, the word
‘Amplitude’ retained a similar appearance in all of the images and was the
only feature visible in the images obtained under $y$-polarization (Fig.
3a-right) and in the absence of the metasurface (Fig. 3c-bottom). Finally, the
image obtained under $x$-polarization was subtracted from that obtained under
$y$-polarization (Fig. 3d) to qualitatively isolate the phase. However, parts
of the word ‘Amplitude’ remained visible in the subtracted image that can be
attributed to some non-uniformity between the two images.

Figure 4: Tunable multi-modal microscopy of biological samples. (a) The output
images of C. elegans (i-iv) and human prostate cancer cells (v-viii). These
include images obtained with $x$-polarized illumination (first column, false-
color), $y$-polarized illumination (second column, false-color), a bright
field microscope (third column, true color) and a DIC microscope (final
column, true color). (b) A schematic of the experimental configuration. (c)
Schematics of bright field (i) and DIC (ii) microscopes. The dashed lines in
the images indicate the regions that are zoomed in the insets.

To demonstrate the potential of the metasurface for bio-imaging applications,
tunable multi-modal microscopy was experimentally performed on samples of C.
elegans (Fig. 4a-i-iv) and unstained human prostate cancer cells (Fig. 4a-v-
viii). The experiments were performed using the configuration illustrated in
Fig. 4b. The steps taken to prepare the samples and details of the
experimental configuration are given in the Methods section. For comparison,
bright field (Fig. 4a-third column) and DIC (Fig. 4a-last column) microscope
images were obtained in the absence of the metasurface using the
configurations illustrated in Fig. 4c.

The semi-transparent body and embryos of the C. elegans were visible in the
image obtained under $x$-polarization (Fig. 4a-i), similar to the DIC
microscope image (Fig. 4a-iv). On the other hand, only the embryos and the
outline of the body were visible in the images obtained under $y$-polarization
(Fig. 4a-ii) and with the bright field microscope (Fig. 4a-iii). Two videos of
the imaging experiments are provided in the Supplementary Information,
including a recording of a live nematode, demonstrating the capacity for real-
time dynamic monitoring. The images of the prostate cancer cells had similar
distinguishing features between the different types of imaging modalities. The
cellular structure and internal details were visible in the image obtained
under $x$-polarization (Fig. 4a-v) and in the DIC microscope image of a
similar region of the sample (Fig. 4a-viii). On the other hand, only some of
the outlines of the cells were visible under $y$-polarization (Fig. 4a-vi) and
in the bright field microscope image (Fig. 4a-vi).

Figure 5: Tunable multi-modal histology of human breast tissue. (a) The images
obtained with the metasurface under $x$\- ((a)-i, false-color) and
$y$-polarization ((a)-ii, false-color). ((b)-i) A bright field microscope
image of the sample stained with H&E obtained under unpolarized white light
illumination (true color). ((b)-ii) The same as (b)-i but for the unstained
sample (true color). ((b)-iii) A DIC microscope image of the unstained sample
obtained under white light illumination (true color). (c) The insets of the
numbered regions in the images in (a)-i and (b)-iii.

Finally, the metasurface was applied to imaging unstained human breast tissue
using the configuration illustrated in Fig. 4b, as before. The samples were
derived from retrospective patients diagnosed with ductal carcinoma in situ
during the course of regular pathology. The steps taken to prepare the samples
are detailed in the Methods section. For comparison, an additional bright
field microscope image was taken of the sample stained with H&E (Fig. 5b-i).

As anticipated, nothing was visible in the images of the unstained sample with
the metasurface under $y$-polarization (Fig. 5a-ii) and with a bright field
microscope (Fig. 5b-ii). The bright field microscope image of the stained
sample (Fig. 5b-ii) displayed clear and color-coded contrast. The image
contained normal breast tissue elements, including background fibrotic stroma,
small capillaries (Fig. 5b-i, feature 3), mature lymphocytes and normal breast
ducts (Fig. 5b-i, features 1 and 2) with a dual cell layer involving myo-
epithelial cells and inner epithelial cells with central secretion material.
Some of these elements can be identified in the images produced by the
metasurface under $x$-polarization (Fig. 5a-i) and a DIC microscope (Fig.
5b-iii). In particular, the simple circular outline of the small capillary can
be identified at feature 3 and three benign breast ducts can be identified as
simple toroidal structures at features 1 and 2. These normal architectural
patterns in tissue can then be distinguished from tumors that often form
aberrant arrangements. However, the capacity to visualize finer architectural
complexity or cellular detail is uncertain, as the region only included
relatively simple benign structures and individual cellular elements could not
be visualized using these methods. The sample was significantly thicker than
the samples in Fig. 4, which provided a challenge for the imaging modalities.
A more ideal application would be the preparation of cytology material from
either a touch or smear preparation, which could provide thinner samples and
allow for assessing individual cellular elements in the images.

## Discussion

Many multi-modal microscopy systems typically involve complicated
configurations due to the use of beam-splitters, fiber bundles or the
multiplexing of wavelengths. While the cascading of imaging systems can
generally be implemented in a laboratory, it can be difficult to integrate
into portable systems. The metasurface in this article could generate bi-modal
images within the same beam and field of view and has the potential to be
integrated into a portable system that is considerably smaller than existing
configurations. It did not require the use of beam-splitters or dichroic
mirrors and was limited only by the speed of polarization switching. The
control over the imaging modality through the polarization is a relatively
straightforward technique given that polarization is an external parameter of
an imaging system. It can be adjusted either mechanically or electronically,
such as through the use of electronic variable retarders. The use of a
polarization-sensitive camera could further enable single-shot capturing of
the bi-modal images.

The image quality produced by the metasurface was similar to that achieved in
DIC microscopy (for $x$-polarization) and bright field microscopy (for
$y$-polarization). In DIC microscopy, the image quality heavily relies on the
optical path length differences not exceeding a specific threshold to avoid
destructive interference that can lead to reduced visibility or artifacts in
regions of strong phase gradients. On the other hand, the metasurface relies
only on the spatial frequencies of the input image overlapping with the
numerical aperture of the metasurface. It is also unlike other Fourier-based
techniques such as Zernike microscopy or Schlieren imaging, which often
struggle to discriminate between positive and negative phase gradients of an
optical field and tend to produce artifacts in images when the phase gradients
are too strong. What sets the metasurface apart is that it could produce
images without the need for contrast-enhancing agents, nor the use of digital
processing or scanning commonly associated with computational techniques such
as the transport of intensity equation [58] or ptychography [59]. However,
phase gradients corresponding to spatial frequency information that is outside
the numerical aperture of the metasurface can be missed, therefore requiring
magnification or a larger numerical aperture.

Dielectric metasurfaces, particularly those with high refractive index
materials such as the silicon in this study, are useful due to their
durability and strong Mie resonances [60]. The metasurface presents
opportunities for integration into existing portable systems, including smart
phones, in combination with LEDs and artificial intelligence to classify
images. The use of meta-lenses can further reduce the overall size of the
system [61]. The simple geometry of the metasurface holds potential for
scalable mass production and can serve as building blocks for more complicated
structures optimized through an inverse design approach. While the relatively
small numerical aperture of the metasurface is advantageous for imaging larger
samples, it can potentially miss finer details or smaller samples without
further magnification. A metasurface with an enhanced or tunable numerical
aperture could overcome this. The anisotropy of the 2D optical transfer
function of the metasurface (Supplementary Information S2.1) was a consequence
of the use of rectangular resonators. Therefore, precise rotations of the
metasurface with respect to the optical axis were required to ensure
consistent performance.

The findings in this article highlight the potential of the metasurface to be
used in a wide range of applications, including real-time biological imaging
and dynamic monitoring [62]. Specifically, it has potential for the early
detection of structural abnormalities in tissue without the requirement for
time-consuming and costly sample processing, nor expensive optical
microscopes. It emerges as a potentially valuable tool for health
professionals that can be used in conjunction with other techniques for point-
of-care testing where access to skilled laboratory labour and specialized
hardware may be unavailable. Finally, some possible extensions to this work
could encompass 2D image processing [63], combining other imaging modalities
such as fluorescence microscopy or polarization-difference imaging, and
hyperspectral imaging due to the sensitivity of the metasurface to the
refractive index of the surrounding.

## Conclusions

In conclusion, we have demonstrated a polarization-tunable, near-infrared
multi-modal microscopy of biological and other samples using a metasurface.
The metasurface exhibited an approximately linear optical transfer function
for polarization along one direction due to the excitation of a Mie resonance,
while maintaining an approximately flat optical transfer function for the
orthogonal polarization. This property enabled the bi-modal switching between
phase contrast and bright field imaging with a switching contrast of up to
$\delta=93\%$. It was experimentally demonstrated across a range of samples,
including C. elegans, prostate cancer cells and breast tissue. Notably, we
distinguished between breast ducts and blood vessels in unstained breast
tissue derived from a patient with breast cancer. Overall, the importance of
these results lies in their capacity to address a gap where other techniques
are either unavailable or require complex sample preparation. It offers an
advance toward the development of compact imaging technology, holding
promising applications in point-of-care testing and remote pathology [64].

### Methods

#### Fabrication

The fabrication of the metasurface started with the deposition of
$236\text{\,}\mathrm{nm}$ of amorphous silicon using plasma-enhanced chemical
vapor deposition (PECVD, Oxford Instruments PLASMALAB 100 PECVD) on a
$23\text{\,}\mathrm{nm}$ film of indium tin oxide (ITO) coating a
$10\text{\,}\mathrm{mm}$ thick glass substrate. This was then spin-coated with
approximately $200\text{\,}\mathrm{nm}$ of poly-methyl methacrylate resist
(MicroChem, PMMA 950A) and electron beam lithography (Vistec EBPG5000plusES)
patterned the structure with a dosage of
$900\text{\,}\mathrm{\SIUnitSymbolMicro C}\text{\,}{\mathrm{cm}}^{-2}$. This
was followed by a step of electron beam evaporation (Intlvac Nanochrome II) of
$10\text{\,}\mathrm{nm}$ of aluminium oxide (Al2O3) and a step of lift-off to
create the hard mask for silicon etching. Pseudo Bosch etching was performed
by inductively coupled plasma - refractive ion etching (ICP-RIE, Oxford
Instruments PlasmaPro System100). The scanning electron microscope images were
taken using a scanning electron microscope (FEI NovaNanoSEM 430). A flowchart
of the fabrication steps is given in the Supplementary Information S4.1.

#### Simulations

Numerical simulations were performed using COMSOL Multiphysics 6.1
implementing the finite element method within the wave optics module. Periodic
Floquet boundary conditions were applied to the sides of the unit cell of the
metasurface to model an infinite 2D lattice. The glass substrate and air
superstrate were assumed to be homogeneous and lossless materials with a
refractive index of 1.5 and 1, respectively. They were both taken to be
$750\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick, which was much thicker than
the silicon resonators. The silicon material was assumed to be a homogeneous
and lossless material with a refractive index of 3.67 at the wavelength of
$\lambda=$969\text{\,}\mathrm{nm}$$. Port boundary conditions were applied
onto the top and bottom of the unit cell to launch and absorb electromagnetic
plane waves, respectively, and to monitor the amplitude and phase of reflected
and transmitted orders. The plane waves had an electric field strength of
$1\text{\,}\mathrm{V}\mathrm{/}\mathrm{m}$ and an incident power of
$1\text{\,}\mathrm{W}$. The finite meshing elements in the model had a maximum
size of $\lambda/$$8\text{\,}\mathrm{nm}$ and a minimum size of
$5\text{\,}\mathrm{nm}$. The optical transfer function of the metasurface was
obtained by computing the complex coefficient $S_{21}$ of the scattering
matrix of the two-port system.

#### Normal-incidence spectroscopy

The metasurface was mounted on an XYZ stage and was illuminated by collimated
white light from a fiber-coupled halogen lamp (Mikropack HL-2000-FHSA) through
a microscope objective (Olympus UPlanFI 4x/0.13). The illumination was
linearly polarized (Thorlabs LPVIS050-MP2 550nm-1.5um) to produce $x$\- or
$y$-polarized light. The light transmitted through the metasurface was
collected by a microscope objective (Nikon ELWD 50x/0.55) and a lens (Thorlabs
lens LA1131-A $f=$50\text{\,}\mathrm{mm}$$). A second lens (Thorlabs lens
LA1951-A $f=$25\text{\,}\mathrm{mm}$$) focused the output light onto the
fiber-port of a spectrometer (Ocean Insight NIR quest). The acquisition time
of each measurement was $1\text{\,}\sec$ and the data was normalized with
respect to the spectra obtained from an unpatterned region of the sample (i.e.
the glass substrate). A beam-splitter (Thorlabs CM1-BS013) was used before the
second lens to relay an image of the metasurface onto a camera (Thorlabs
DCC1645C-HQ) using a lens (Thorlabs lens LA1608-B f = 75 mm). A schematic of
the experimental configuration is given in the Supplementary Information S3.2.

#### Angle-dependent spectroscopy

The angle-dependent spectroscopy experiments were performed using a back focal
plane imaging configuration. The metasurface was illuminated with a broadband
tungsten halogen white light source (Ocean Optics HL-2000-HP) through an air
microscope objective (Olympus 10x/0.25). The metasurface was mounted on the
stage of an inverted microscope (Olympus IX71). The illumination was linearly
polarized (Thorlabs LPVIS100) to produce $x$\- or $y$-polarized light. It was
then expanded to fill the microscope objective back aperture to excite the
metasurface with the full angular range as determined by the numerical
aperture of the objective. The reflected light over the same angular range was
collected by the same microscope objective and an image of the back focal
plane was relayed to a spectrograph (Acton SP300i)/CCD camera (Princeton Pixis
1024BR) couple by a tube lens (Olympus single port tube lens
$f=$180\text{\,}\mathrm{mm}$$) and Bertrand lens (Thorlabs TTL180-A
$f=$180\text{\,}\mathrm{mm}$$) pair. A second polarizer (Thorlabs LPVIS100)
selected the co- or cross-polarized components of the output. One axis of the
back focal plane image was transformed into direction by the spectrograph to
provide an angle-resolved reflectance distribution about the wavelength of
$969\text{\,}\mathrm{nm}$. The acquisition time of each measurement was
$90\text{\,}\sec$ and the data was normalized with respect to the data
obtained using a mirror in place of the metasurface. The normalized
reflectance data was converted to transmittance data by $T=1-R-A$, where the
absorptance $A$ of the metasurface was calculated to be approximately zero in
the Supplementary Information S2.1-2.2. The transmission amplitude was then
obtained by taking the square root of the transmittance data. A schematic of
the experimental configuration is given in the Supplementary Information S3.2.

#### Multi-modal microscopy - spatial light modulator

The imaging experiments shown in Fig. 3 were performed using light from a
super-continuum source (NKT Photonics SuperK COMPACT laser). The wavelength
was controllable through the NKT Photonics Control software and a tunable
multi-channel filter (NKT SuperK SELECT). The power of the light was measured
to be $0.02\text{\,}\mathrm{mW}$ by a power meter (Thorlabs PM1000D). The
illumination was linearly polarized (Thorlabs LPNIR050-MP2) to be compatible
with the SLM (Holoeye Pluto-VIS-001 LCOS-SLM). The reflected light was
expanded and collimated by a microscope objective (Olympus Plan N 20x/0.40)
and two lenses (Thorlabs LA1027-B $f=$35\text{\,}\mathrm{mm}$$ and LA1509-B
$f=$100\text{\,}\mathrm{mm}$$) to fill the window range of the SLM. The
amplitude imaging mask was mounted directly in front of the SLM and the SLM
replicated phase profiles from a computer with a phase excursion of $\pi$
radians. A half wave-plate (Thorlabs WPH05M-1064) was used to produce either
$x$\- or $y$-polarized light. A lens (Thorlabs LA1433-B
$f=$150\text{\,}\mathrm{mm}$$) and microscope objective (Nikon LU Plan
20x/0.4) projected a de-magnified image of the SLM onto the metasurface. The
metasurface was tilted by $2\text{\,}\mathrm{\SIUnitSymbolDegree}$ using an
XYZ rotation stage. A microscope objective (Olympus Plan N 20x/0.4) and a lens
(Thorlabs LA1986-B $f=$125\text{\,}\mathrm{mm}$$) relayed the magnified output
image onto a camera (Andor Zyla 4.2P sCMOS). The exposure time of the camera
was $0.06\text{\,}\sec$. A schematic of the experimental configuration is
given in the Supplementary Information S3.3.

The amplitude imaging mask was $50\text{\,}\mathrm{mm}$ by
$40\text{\,}\mathrm{mm}$ in size. It was fabricated on a
$2.5\text{\,}\mathrm{mm}$ thick soda lime glass photo-mask plate with
$150\text{\,}\mathrm{nm}$ of chrome (Cr) and an anti-reflective coating. The
plate was coated with AZ1518 photo-resist and it was exposed to ultra-violet
light with a wavelength of $365\text{\,}\mathrm{nm}$ (IMP600 direct write
system, dose $75\text{\,}\mathrm{mJ}\text{\,}{\mathrm{cm}}^{-2}$) to define
the pattern. The exposed photo-resist was removed by developing in AZ400K
developer diluted in water (4 parts water to 1 part AZ400K) for
$1\text{\,}\mathrm{min}$. It was then rinsed with de-ionized water and dried
with nitrogen gas (N2). The exposed chrome was removed by immersing the plate
in a chrome etchant (TechniEtch CR01) for $60\text{\,}\sec$ and then it was
rinsed with de-ionized water and dried with nitrogen gas (N2). The remaining
resist was removed by exposing it to ultra-violet light with a wavelength of
$365\text{\,}\mathrm{nm}$ (IMP600 direct write system, dose
$150\text{\,}\mathrm{mJ}\text{\,}{\mathrm{cm}}^{-2}$) and developing it in
AZ400K developer diluted with water.

#### Multi-modal microscopy - biological samples and polarization switching

The biological imaging experiments (Figs. 4 and 5) were performed using light
from a super-continuum source (NKT Photonics SuperK COMPACT and NKT Photonics
SuperK SELECT), as above. It was linearly polarized (Thorlabs LPNIR050-MP2) to
produce $x$\- or $y$-polarized light. This illuminated the biological sample
that was mounted on an XYZ sample holder (Thorlabs SFH3 fixed filter holder).
A half wave-plate (Thorlabs WPH05M-1064) was used to control the polarization.
The metasurface was placed immediately after the sample and it was tilted by
$2\text{\,}\mathrm{\SIUnitSymbolDegree}$ by an XYZ rotation stage. A
microscope objective (Nikon LU Plan 20x/0.4) and a lens (Thorlabs LA1986-B
$f=$125\text{\,}\mathrm{mm}$$) relayed the output image onto a camera (Andor
Zyla 4.2P sCMOS). A schematic of the experimental configuration is given in
the Supplementary Information S3.4.

The polarization switching experiment (Fig. 3d) was performed with the same
configuration. In this case, the metasurface was illuminated with $x$\- or
$y$-polarized light at normal incidence and there was no imaging sample placed
in the experiment. The output transmittance through the metasurface was
measured using the camera. The transmittance was normalized to that measured
in the absence of the metasurface. The measurements were repeated for
different states of linear polarization that was controlled by the half wave-
plate.

#### Sample preparation

C. elegans. The C. elegans were wild type (stain N2) animals and were obtained
from the Caenorhabditis Genetics Center at the University of Minnesota. The
cultures were maintained at $20\text{\,}\mathrm{\SIUnitSymbolCelsius}$ using
standard protocols on nematode growth media (NGM) supplement with E. coli
(strain OP50) [65]. Developmentally synchronous animals that were derived from
eggs laid over a $2\text{\,}\mathrm{h}$ window, and then cultured for an
additional $72\text{\,}\mathrm{h}$, were used for imaging. These young adult
worms were isolated, mounted under glass cover-slips on 2% (w/v) agarose pads,
and immobilized via brief heat shock of
$60\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for $10\text{\,}\sec$ prior to
imaging.

Prostate cancer cells. The human prostate cancer cells (PC-3) were obtained
from the American Type Cell Collection (ATCC, USA). The cells were seeded on a
glass coverslip at a seeding density of 300,000 cells. The cells were then
grown in complete RPMI media (10% FBS, 1% Penicillin/streptomycin) for
$24\text{\,}\mathrm{h}$ in a humidified incubator at
$37\text{\,}\mathrm{\SIUnitSymbolCelsius}$ with 5% carbon dioxide (CO2). The
cells were then washed with Dulbecco’s phosphate-buffered saline (DPBS) and
fixed with 4% paraformaldehyde. The glass coverslip with the cells was mounted
on a microscope slide. The RPMI 1640 Culture medium, fetal bovine serum,
Penicillin/streptomycin antibiotic, Dulbecco’s phosphate-buffered saline (no
calcium, no magnesium) and 16% methanol free paraformaldehyde, were all
procured from ThermoFisher Scientific, Australia.

Breast tissue. The human breast tissue samples were obtained surgically
through the course of regular pathology and were prepared at Dorevitch
Pathology in Melbourne. The samples were obtained with informed consent from
the patients that was approved by the University of Melbourne ethics approval
(project ID 27303). The tissue was fixed with formalin and contained known
invasive breast tumors, for which all clinical diagnostic sampling and formal
reporting had been completed. The tissue samples were approximately
3-$4\text{\,}\mathrm{mm}$ thick and were taken from benign and tumor-involved
regions of the breast. These sections underwent standard laboratory tissue
processing with paraffin infusion on the Evident VIP6 platform. The samples
were then embedded within paraffin into tissue blocks and cut via microtomy
into sections that were $3\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in
thickness. These tissue slices were then placed onto charged microscope
slides. One of the tissue sections was stained with haematoxylin and eosin
staining profile and a coverslip was applied.

#### Conventional microscopy

Bright field and DIC microscopy were performed using an inverted microscope
(Olympus BX60) with white light illumination, a microscope objective (Olympus
U Plan FI 20x/0.5) and a camera (Nikon Z5). A green filter (Grubb Parsons
$590\text{\,}\mathrm{nm}$ filter) was used in DIC microscopy to enhance the
contrast of the output image.

Supplementary information

Supplementary material is available in the Supplementary Information. This
includes one Supplementary Information document and four Supplementary videos.

Authors’ contributions

A.R, S.B.S and L.W conceptualized the project. The idea of near-infrared phase
microscopy with a silicon-on-glass metasurface was put forward by A.R. S.B.S
designed the metasurface and put forward the idea of polarization-tunable
multi-modal microscopy with L.W. S.B.S conducted the theoretical and
computational work. The metasurface was fabricated by J.M at the Melbourne
Centre for Nanofabrication, after preliminary versions of the metasurface were
fabricated by L.W. N.P built the optical setup for normal-incidence
spectroscopy and J.A.H built the optical configuration for angle-dependent
spectroscopy. S.B.S collected the normal-incidence spectroscopy data and also
the angle-dependent spectroscopy data with assistance from J.A.H. S.B.S built
the optical configuration for imaging and performed the imaging experiments.
W.S prepared the prostate cancer samples and G.M prepared the C. elegans
samples. K.R obtained the breast tissue specimens, which were prepared as
fixed pathology samples by M.M and Dorevitch Pathology in Melbourne. M.M
provided a pathology analysis of the breast tissue images and contributed to
the discussion of the images. S.B.S processed the data, created the figures,
wrote the article and prepared the manuscript. All authors provided input on
the manuscript. A.R, L.W, K.B.C and A.A.S supervised the project.

Acknowledgments

This work was performed in part at the Melbourne Centre for Nanofabrication
(MCN) in the Victorian node of the Australian National Fabrication Facility
(ANFF). Dan Smith from MCN is acknowledged for preparing the imaging target
used in the experiments involving the spatial light modulator. Wendy S. L. Lee
from the University of New South Wales and Dragomir Neshev from the Australian
National University are acknowledged for their support in preliminary
spectroscopy measurements. Rajour T. Ako from RMIT University is acknowledged
for contributing to the etching process in the fabrication of a preliminary
version of the metasurface. Ravi Shukla, also from RMIT University, is
acknowledged for providing PC-3 cell culture resources. The authors further
acknowledge Timothy J. Davis, Timur Gureyev, Snjezana Tomljenovic-Hanic,
Sivacarendran Balendhran and Benjamin Russell from the University of Melbourne
for their useful discussions.

### Declarations

Funding

This research was funded by the Australian Government through the Australian
Research Council (ARC) Centre of Excellence grant CE200100010. S.B.S
acknowledges the support of the Ernst & Grace Matthaei Scholarship and the
Australian Government Research Training Program Scholarship. J.A.H
acknowledges the ARC projects CE170100026 and FT180100295. N.P acknowledges
the support of a Melbourne Research Scholarship.

Competing interests

The authors declare no competing interests.

Ethics approval

Ethics approval and informed patient consent for the breast tissue samples
were approved by the University of Melbourne Human Research Ethics Committee
under the application identification number 27303.

Availability of data and materials

The data that support the findings within this paper are available from the
corresponding authors upon reasonable request.

Code availability

The computer code that support the findings within this paper is available
from the corresponding authors upon reasonable request.

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Metasurfaces for infrared multi-modal microscopy: phase contrast and bright
field - supplementary information

###### Contents

  1. S1 Theory
    1. S1.1 Image processing
    2. S1.2 Multipole decomposition
    3. S1.3 Mie scattering of an isolated resonator
  2. S2 Additional simulation data
    1. S2.1 Optical response
    2. S2.2 Performance metrics
    3. S2.3 Isolated resonator
    4. S2.4 Imaging simulations
  3. S3 Optical configurations
    1. S3.1 Polarization
    2. S3.2 Spectrometry
    3. S3.3 Imaging with a spatial light modulator
    4. S3.4 Biological microscopy
  4. S4 Additional experimental data
    1. S4.1 Metasurface fabrication
    2. S4.2 Optical characterization
    3. S4.3 Performance metrics
    4. S4.4 Imaging experiments
    5. S4.5 Biological imaging
    6. S4.6 Experimental videos

##  S1 Theory

###  S1.1 Image processing

The fundamental concepts for the application of the metasurface to tunable
multi-modal microscopy are briefly described in the main article. Here, the
explanation is expanded to explain how the metasurface operates in the
formation of images. Consider a source of monochromatic plane waves with an
electric field $\vec{E}_{s}(x,y,z)$, wavelength $\lambda$ and wave-vector
$\vec{k}$ propagating along the $z$-axis. Assuming polarized light and
ignoring vector effects for the time being, suppose that this source of light
illuminates a semi-transparent sample located in the $z=0$ plane with a
transmission function $O(x,y)=O_{0}(x,y)e^{i\varphi(x,y)}$ in the Kirchhoff
approximation. Here, $O_{0}(x,y)$ represents the amplitude features of the
sample and $\varphi(x,y)$ represents the phase features. This defines an
optical image in the $z=0$ plane with an intensity profile given by
$I_{0}(x,y)=|O(x,y)|^{2}=|O_{0}(x,y)|^{2}$, which represents the bright field
image of the sample. Visualizing the phase features of the sample requires
highlighting the phase shifts imparted onto the light transmitted through the
sample, which can be modelled as,

$\vec{E}_{\text{in}}(x,y)=O_{0}(x,y)e^{i\varphi(x,y)}\vec{E}_{s}(x,y)\ .$ (S4)

The metasurface filters the spatial frequency components of an image in either
the image plane of the sample or directly in the object plane. The optical
response of the metasurface in $k$-space is characterized by the optical
transfer function, which is defined as the Fourier transform of its Green’s
function. In general, the optical transfer function is a rank-2 tensor (to
accommodate polarization) given in the $\\{p,s\\}$-polarization basis as,

$H(k_{x},k_{y})=\begin{pmatrix}H_{pp}(k_{x},k_{y})&H_{ps}(k_{x},k_{y})\\\
H_{sp}(k_{x},k_{y})&H_{ss}(k_{x},k_{y})\\\ \end{pmatrix}\ .$ (S5)

The spatial frequencies $k_{x}$, $k_{y}$ and
$k_{z}=\sqrt{|\vec{k}|^{2}-k_{x}^{2}-k_{y}^{2}}$ are the Cartesian projections
of $\vec{k}$, where $k_{0}=|\vec{k}|=2\pi/\lambda$ is the wavenumber. As in
Ref. [1], the spatial frequency components can be represented as a function of
the spherical coordinates $(r=k_{0},\theta,\phi)$,

$\begin{gathered}k_{x}=k_{0}\sin(\theta)\cos(\phi)\\\
k_{y}=k_{0}\sin(\theta)\sin(\phi)\\\ k_{z}=k_{0}\cos(\theta)\ ,\end{gathered}$
(S6)

where $(\theta,\phi)$ are the polar and azimuthal propagation angles of the
plane waves with respect to the $z$-axis, respectively. Any light beam can be
decomposed into bundles of weighted plane waves, each travelling in different
directions, by the Fourier transform [2],

$\mathscr{F}\\{E(x,y)\\}(k_{x},k_{y})=\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}E(x,y)e^{-i(k_{x}x+k_{y}y)}dxdy\
,$ (S7)

where $\mathscr{F}$ is the Fourier transform operator. Also, given that the
optical transfer function represents the response of the metasurface to plane
waves in $k$-space, then it follows that the spatial frequency dependence of
the metasurface in $k$-space is equivalent to an angular dependence in real
space. This is the concept that is fundamental to the design of the
metasurface.

The impact of the metasurface on an input image is determined by the Green’s
function of the metasurface. The output image is given by a convolution
integral between the input image and the Green’s function,

$E_{i}(x,y)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}G_{ij}(x-x^{\prime},y-y^{\prime})E_{j}^{\text{in}}(x^{\prime},y^{\prime})dx^{\prime}dy^{\prime}\
,$ (S8)

where $G(x,y)$ is the Green’s function tensor and $E_{i}$ is the
$i^{th}$-component of the output electric field. However, by the convolution
theorem, the impact of the metasurface on the spatial frequency spectrum of
the input image can be directly modelled as,

$\tilde{E}_{i}(k_{x},k_{y})=H_{ij}(k_{x},k_{y})\tilde{E}^{\text{in}}_{j}(k_{x},k_{y})\
,$ (S9)

where $\tilde{E}=\mathscr{F}\\{E\\}$. The convolution theorem simplifies the
relationship between the input image and the output image from a convolution
in real space to a multiplication with the optical transfer function in
$k$-space. The output electric field $E$ can be recovered from $\tilde{E}$ by
the inverse Fourier transform,

$\mathscr{F}^{-1}\\{\tilde{E}(k_{x},k_{y})\\}(x,y)=\frac{1}{2\pi}\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}\tilde{E}(k_{x},k_{y})e^{i(k_{x}x+k_{y}y)}dk_{x}dk_{y}\
,$ (S10)

where $\mathscr{F}^{-1}$ is the inverse Fourier transform operator and
$E=\mathscr{F}^{-1}\\{\tilde{E}\\}$. Therefore, given the following property
for the Fourier transform of a derivative,

$\mathscr{F}\left\\{\frac{\partial E}{\partial
x}\right\\}(k_{x},k_{y})=ik_{x}\tilde{E}(k_{x},k_{y})\ ,$ (S11)

then a linear optical transfer function, $H\propto k_{x}$, can produce the
spatial derivative (up to a multiplicative constant) of an incoming light
field along the $x$-direction. It follows that a metasurface with an
approximately linear angular dispersive transmission
$T(\theta)\propto\sin(\theta)\approx\theta$ (for $\phi=0$ and small values of
$\theta$) is capable of phase visualization in the object or image plane of a
sample. Here, the {$p$, $s$}-polarization basis was used but, in general, the
optical transfer function tensor can be expressed in any polarization basis.
Henceforth, we utilize an $\\{x,y\\}$ basis, i.e. $H_{xx}$, $H_{xy}$, $H_{yx}$
and $H_{yy}$, describing the impact of the metasurface on the transverse
components of the incident field. It was shown in the main article that if we
let the $x$-axis be aligned with the long axis of the silicon blocks for an
un-tilted metasurface (normal incidence), then the metasurface has an
approximately linear optical transfer function at a tilt of
$2\text{\,}\mathrm{\SIUnitSymbolDegree}$ about the $y$\- (short) axis for
$x$-polarized illumination, i.e. $H_{xx}\propto k_{x}+k_{x0}$, where $k_{x0}$
is a constant. This can compute the derivative along the $x$-direction, plus a
constant offset that pertains to a non-zero background in the output image.
Hence, the design of the metasurface was carried out by optimizing the angular
response of the metasurface in real space to engineer its optical transfer
function in $k$-space.

In the case of a pure phase sample that has $|O_{0}(x,y)|=\text{constant}$,
then the sample is visualized because the phase derivative is converted into
intensity variations in the output image $I(x,y)$,

$I(x,y)\approx\left|O_{0}\left(\frac{\partial\varphi(x,y)}{\partial
x}+k_{x0}\right)\right|^{2}\ .$ (S12)

This result is obtained when a linear optical transfer function
($H_{xx}\propto k_{x}+k_{x0}$) and Eq. (S4) are substituted into Eq. (S9).
Hence, the output image is proportional to $|\partial\varphi(x,y)/\partial
x|^{2}$, with contrast created in regions where the phase is varying along the
direction of differentiation. The constant offset in the optical transfer
function is due to the metasurface being slightly tilted by
$2\text{\,}\mathrm{\SIUnitSymbolDegree}$, corresponding to a normalized
spatial frequency of $k_{x0}/k_{0}\approx 0.03$, that shifts the $k$-space
origin about the angular offset. The increased transmission level at this
offset preserves some of the illumination to create a non-zero background in
the output image. This distinguishes the imaging modality from dark field
imaging that has a dark background in the output image. One could also
uniquely distinguish the phase derivatives as they will manifest as different
grayscale levels that are above or below the background, corresponding to the
region of zero phase gradient, i.e. $I(x,y)\approx|O_{0}k_{x0}|^{2}$.

On the other hand, the metasurface was approximately transparent when the
illumination was $y$-polarized. This corresponds to a flat optical transfer
function, $H_{yy}\approx\text{constant}$, that only attenuates the input
image. In the case that the constant is equal to one, the output image will be
exactly the input image. As a result, the output image is proportional to the
bright field image, i.e. $I(x,y)\propto|O(x,y)|^{2}$. The cross-polarized
components of the optical transfer function $H_{xy}$ and $H_{yx}$ (Fig. S7 and
Table 1) are zero in the region of operation. However, they have some non-zero
contributions for larger values of $k_{x}$ and $k_{y}$, but these
contributions were measured to be negligible in the experiments by using an
analyzing polarizer on the output image.

The performance of the polarization switching was quantified by the switching
contrast of the metasurface, defined as $\delta=H_{yy}(0,0)-H_{xx}(0,0)$. A
similar value can be defined for the switching contrast of the transmittance,
given by $\Delta=T_{y}-T_{x}$, where $T_{x}$ and $T_{y}$ are the transmittance
levels under $x$\- and $y$-polarized illumination at the operating wavelength,
respectively. The efficiency of the metasurface for image processing is
defined as $\eta=|H_{xx}(k_{x}^{\text{max}},0)|$, where $k_{x}^{\text{max}}$
is the maximum spatial frequency component where the optical transfer function
of the metasurface is still within its numerical aperture. The numerical
aperture of the metasurface, NA, is defined to be the width of the optical
transfer function where the above conditions are met. This is contrasted from
the numerical aperture of the imaging system itself, which is determined by
the optical components in the system.

###  S1.2 Multipole decomposition

Fig. 2a-iii of the main article shows a multipole decomposition of the
transmittance spectrum of the metasurface. The method from Ref. [3] was used
to do this and it is briefly summarized here. Consider a metasurface composed
of an array of identical dielectric resonators. For normally incident,
$x$-polarized illumination, the multipole decomposition of the transmittance
of the metasurface up to second order is given by,

$T=\left|1+\frac{ik}{2E_{0}A_{\text{cell}}\epsilon_{0}\epsilon_{s}}\left(p_{x}+\frac{1}{v_{s}}m_{y}-\frac{ik}{6}Q_{xz}-\frac{ik}{2v_{s}}M_{yz}+\cdots\right)\right|^{2}\
.$ (S13)

Here, $k=2\pi/\lambda$ is the wave-number in the surrounding, $E_{0}$ is the
magnitude of the incident electric field, $\lambda$ is the wavelength of the
illumination, $A_{\text{cell}}$ is the area of the lattice unit cell,
$\epsilon_{s}$ is the relative permittivity of the surrounding and $v_{s}$ is
the speed of light in the surrounding. Furthermore, $\vec{p}$ is the electric
dipole moment vector, $\vec{m}$ is the magnetic dipole moment vector, $Q$ is
the electric quadrupole moment tensor and $M$ is the magnetic quadrupole
moment tensor. The relevant components of the multipole moments are given by,

$\begin{gathered}p_{x}=\int
P_{x}j_{0}(kr)dV+\frac{k^{2}}{10}\int\frac{15j_{2}(kr)}{(kr)^{2}}\left((\vec{r}\cdot\vec{P})x-\frac{1}{3}r^{2}P_{x}\right)dV\\\
m_{y}=-\frac{i\omega}{2}\int\frac{3j_{1}(kr)}{kr}\left(\vec{r}\times\vec{P}\right)_{y}dV\\\
Q_{xz}=\int\frac{9j_{1}(kr)}{kr}\left(xP_{z}+P_{x}z\right)dV+6k^{2}\int\frac{j_{3}(kr)}{(kr)^{3}}\left(5xz(\vec{r}\cdot\vec{P})-(xP_{z}+P_{x}z)r^{2}\right)dV\\\
M_{yz}=\frac{\omega}{3i}\int\frac{15j_{2}(kr)}{(kr)^{2}}\left((\vec{r}\times\vec{P})_{y}z+y(\vec{r}\times\vec{P})_{z}\right)dV\
,\end{gathered}$ (S14)

where $\vec{r}=(x,y,z)$ is the position vector, $j_{n}$ is the $n^{th}$-order
spherical Bessel function of the first kind,
$\vec{P}(\vec{r})=\epsilon_{0}(\epsilon_{s}-1)\vec{E}(\vec{r})$ is the
polarization density and $\omega=2\pi f$ is the angular frequency. The
integrals are performed over the volume $V$ of the resonator within a single
unit cell of the metasurface. This formalism was set up in COMSOL Multiphysics
6.1 to obtain the multipole decomposition of the transmission spectrum of the
metasurface.

###  S1.3 Mie scattering of an isolated resonator

In 1908, Gustav Mie [4] exactly solved Maxwell’s equations for the scattering
of plane waves from a homogeneous sphere of size comparable to a wavelength.
Although the analytic solution exists only for ellipsoidal particles, the
physics is identical for arbitrary geometries [5]. Consider the Mie scattering
of plane waves by a rectangular resonator. To characterize its multipolar
behaviour, the electric field scattered by the resonator when under plane wave
illumination is expanded in the basis of vector spherical harmonics,

$\vec{E}_{sc}(r,\theta,\phi)=\sum_{n=1}^{\infty}\sum_{m=-n}^{+n}c_{nm}\left(a_{nm}\vec{N}_{nm}(r,\theta,\phi)+b_{nm}\vec{M}_{nm}(r,\theta,\phi)\right)\
,$ (S15)

where $a_{nm}$ and $b_{nm}$ are the electric and magnetic scattering
coefficients, respectively. The normalization constants are defined as,

$c_{nm}=|\vec{E}_{0}|i^{n+2m-1}\sqrt{\frac{(2n+1)(n-m)!}{4(n+m)!}}\ ,$ (S16)

where $\vec{E}_{0}$ is the electric field of the illumination. Following the
formalism from Ref. [6], the electric and magnetic harmonics are respectively
defined in spherical coordinates as,

$\begin{gathered}\vec{N}_{nm}=\frac{1}{k_{0}r}\biggl{(}\tau_{nm}(\cos(\theta))\hat{\theta}+i\pi_{nm}(\cos(\theta))\hat{\phi}\biggr{)}e^{im\phi}\frac{\partial}{\partial
r}\biggl{(}rh_{n}^{(1)}(k_{0}r)\biggr{)}\\\
+\frac{n(n+1)}{k_{0}r}P_{n}^{m}(\cos(\theta))h_{n}^{(1)}(k_{0}r)e^{im\phi}\hat{r}\\\
\vec{M}_{nm}=h_{n}^{(1)}(k_{0}r)e^{im\phi}\biggl{(}i\pi_{nm}(\cos(\theta))\hat{\theta}-\tau_{nm}(\cos(\theta))\hat{\phi}\biggr{)}\
,\end{gathered}$ (S17)

where $h_{n}^{(1)}$ is the $n^{th}$-order spherical Hankel function of the
first-kind and $P_{n}^{m}$ is the $n^{th}$-order associated Legendre
polynomial of degree $m$. The $\tau$ and $\pi$ functions are defined as,

$\begin{gathered}\tau_{nm}(\theta)=\frac{\partial}{\partial\theta}P_{n}^{m}(\cos(\theta))\\\
\pi_{nm}(\theta)=mP_{n}^{m}(\cos(\theta))\csc(\theta)\ .\end{gathered}$ (S18)

The scattering coefficients can be computed by projecting the scattered
electric field onto the vector spherical harmonics on a virtual sphere of
radius $r_{0}$ surrounding the resonator, given by,

$\begin{gathered}a_{nm}=\frac{\int_{0}^{2\pi}\int_{0}^{\pi}\vec{E}_{sc}(r=r_{0},\theta,\phi)\cdot\vec{N}^{*}_{nm}(r=r_{0},\theta,\phi)\sin(\theta)d\theta
d\phi}{c_{nm}\int_{0}^{2\pi}\int_{0}^{\pi}|\vec{N}_{nm}(r=r_{0},\theta,\phi)|^{2}\sin(\theta)d\theta
d\phi}\\\
b_{nm}=\frac{\int_{0}^{2\pi}\int_{0}^{\pi}\vec{E}_{sc}(r=r_{0},\theta,\phi)\cdot\vec{M}^{*}_{nm}(r=r_{0},\theta,\phi)\sin(\theta)d\theta
d\phi}{c_{nm}\int_{0}^{2\pi}\int_{0}^{\pi}|\vec{M}_{nm}(r=r_{0},\theta,\phi)|^{2}\sin(\theta)d\theta
d\phi}\ .\end{gathered}$ (S19)

By using the asymptotic relation
$h_{n}^{(1)}(k_{0}r)\sim(-i)^{n}e^{ik_{0}r}/ik_{0}r$ for $k_{0}r\gg 1$ and the
orthonormality of the associated Legendre polynomials, it follows that the
scattering cross section is given by

$\sigma_{s}=\frac{\pi}{k_{0}^{2}}\sum_{n=1}^{\infty}\sum_{m=-n}^{+n}n(n+1)\left(|a_{nm}|^{2}+|b_{nm}|^{2}\right)\
.$ (S20)

The integer $n\in[1,\infty)$ provides the degree of the multipole, i.e. $n=1$
corresponds to dipole, $n=2$ to quadrupole, etc., while the integer
$m\in[-n,n]$ provides their orientations in space. The contribution to the
scattering cross section for each $n$ represents the cross section for that
particular degree of the multipole. Furthermore, for each multipole degree,
the electric component corresponds to the $a_{nm}$ term and the magnetic
component to the $b_{nm}$ term. Hence, the total cross section can be
decomposed as [7],

$\sigma_{s}=\sigma_{s}^{ED}+\sigma_{s}^{MD}+\sigma_{s}^{EQ}+\dotsm\ ,$ (S21)

where ED, MD and EQ each stand for electric dipole, magnetic dipole and
electric quadrupole, respectively. The Mie resonances of the resonator
manifest as peaks in the cross sections at the resonant wavelengths. This
formalism was set up in COMSOL Multiphysics 6.1 to determine the multipolar
decomposition of the scattering cross section for an isolated resonator from
the metasurface.

##  S2 Additional simulation data

###  S2.1 Optical response

Simulations of the optical response of the metasurface were performed using
the steps outlined in the Methods section of the main article. The metasurface
was designed to have an angle-dependent transmission for $x$-polarized
illumination at the operating wavelength, and an angle-independent
transmission for $y$-polarized illumination. The angle-dependent transmission
spectra are given in Fig. S1 as a function of the angle of incidence. The
results show that the transmission spectrum changed under $x$-polarized
illumination with a varying angle of incidence. Meanwhile, the transmission
spectrum under $y$-polarized illumination remained similar when the angle of
incidence was varied, with only the appearance of dark modes for shorter
wavelengths.

Figure S1: Angle-dependent transmission spectra. (a - left) A 2D heat map of
the transmission versus the angle of incidence and the wavelength of
$x$-polarized illumination. (a - right) Same as the left but for
$y$-polarization. (b - left) The 1D transmission spectra for 0, 5 and 10
degrees angle of incidence for $x$-polarized illumination. (b - right) Same as
the left but for $y$-polarized illumination.

The 1D plots of the optical response are given in Fig. S2. The reflectance of
the metasurface is shown to be a maximum at the operating wavelength for
$x$-polarized illumination and a minimum for $y$-polarized illumination (Fig.
S2a). The absorptance is also shown to be nearly zero for both cases. The
transmission spectra for different states of linear polarization shows how the
spectrum morphs between those for $x$\- and $y$-polarized illumination (Fig.
S2b).

Figure S2: 1D spectra as a function of various parameters. (a) The reflectance
and absorptance spectra of the metasurface for $x$\- and $y$-polarized
illumination. (b) The transmittance spectra as a function of the angle of
linear polarization.

The operating wavelength of the metasurface is dependent on the geometrical
parameters of the metasurface, including the size of the resonators (length L,
width W and height H) and the lattice constant (P) of the array. The
dispersion of the resonance as a function of the geometrical parameters is
shown in Fig. S3. In each case, the resonance red-shifted as the geometrical
parameters were increased.

Figure S3: Dispersion of the resonance. (a) A 2D heat map of the transmittance
spectrum as a function of the length of the resonators. (b) Same as a but for
the width of the resonators. (c) Same as a but for the height of the
resonators. (d) Same as a but for the lattice constant of the metasurface. In
all cases, the illumination was $x$-polarized.

The electromagnetic behavior of the metasurface when on-resonance is depicted
in Figs. S4-S6. They each show the electric field, magnetic field and power
flow in the unit cell of the metasurface at the operating wavelength for $x$\-
and $y$-polarized illumination. The electric and magnetic fields when on-
resonance from $x$-polarized illumination display behavior that is linked to
an electric dipole resonance, with the power of the fields swirling around the
resonator. On the other hand, the fields behave differently when off-resonance
and the power of the fields flows through the resonator.

Figure S4: Electromagnetic behaviour of the metasurface in the $xz$-plane.
(a),(d) The electric field strength (heat map) and direction (arrows) in the
unit cell for $x$\- (a) and $y$-polarization (d). (b),(e) Same as (a),(d) but
for the magnetic field. (c),(f) The time-averaged electromagnetic energy
density (heat map) and Poynting vector (arrows) in the unit cell for $x$\- (c)
and $y$-polarization (f). All of the images are given in the $xz$-plane of the
metasurface and at the operating wavelength. Figure S5: Electromagnetic
behaviour of the metasurface in the $yz$-plane. (a),(d) The electric field
strength (heat map) and direction (arrows) in the unit cell for $x$\- (a) and
$y$-polarization (d). (b),(e) Same as (a),(d) but for the magnetic field.
(c),(f) The time-averaged electromagnetic energy density (heat map) and
Poynting vector (arrows) in the unit cell for $x$\- (c) and $y$-polarization
(f). All of the images are given in the $yz$-plane of the metasurface and at
the operating wavelength. Figure S6: Electromagnetic behaviour of the
metasurface in the $xy$-plane. (a),(d) The electric field strength (heat map)
and direction (arrows) in the unit cell for $x$\- (a) and $y$-polarization
(d). (b),(e) Same as (a),(d) but for the magnetic field. (c),(f) The time-
averaged electromagnetic energy density (heat map) and Poynting vector
(arrows) in the unit cell for $x$\- (c) and $y$-polarization (f). All of the
images are given in the $xy$-plane of the metasurface and at the operating
wavelength.

The magnitude and phase of the optical transfer function are given in Fig. S7.
It was computed as the angle-dependent transmission as a function of the polar
and azimuthal angles of incidence ($\theta,\phi)$. The 1D slices through
$k_{x}=0$ and $k_{y}=0$ are given in Fig. S8. The results show that the
capability of the metasurface to process input images along $k_{y}$ is weaker
than that for $k_{x}$ under $x$-polarization. Moreover, the cross-polarized
components of the optical transfer function are consistent with Ref. [8]. They
are non-zero along the diagonals in Fig. S7(middle column), however the
contribution to the output was measured to be small in the experiments.

Figure S7: 2D optical transfer function of the metasurface. The magnitude
(first two columns) and phase (last column) of the optical transfer function
tensor components at the operating wavelength. Figure S8: 1D optical transfer
function of the metasurface. (a) The magnitude and phase of the co-polarized
components of the optical transfer function along $k_{y}=0$. (b) Same as a but
along $k_{x}=0$.

###  S2.2 Performance metrics

The performance of the metasurface was quantified through various numerical
metrics. These included the transmittance, reflectance, absorptance, quality
factor, electric field enhancement, numerical aperture, angular response,
polarization conversion efficiency, processing efficiency and the switching
contrast. The results are tabulated in Table 1 and can be compared to the
experimental results in Table 2. The results show that the metasurface
strongly reflects light when on-resonance with little absorption for normally
incident, $x$-polarized illumination. The electric dipole resonance had a
quality factor of 17.3, which produced an optical transfer function with a
numerical aperture of approximately 0.17.

Table 1: Simulated performance metrics of the metasurface. The performance metrics were obtained through simulations in COMSOL Multiphysics 6.1 at the operating wavelength of $969\text{\,}\mathrm{nm}$, corresponding to the electric dipole resonance of the metasurface. Here, $\vec{E}_{0}$ denotes the electric field of the normally-incident, $x$-polarized illumination. The values $k_{x}^{\text{max}}$ and $k_{x}^{\text{min}}$ denote the spatial frequencies at the edges of the contrast zone. $E_{t}$ and $E_{r}$ denote the electric field transmitted through, and reflected from, the metasurface, respectively. $P_{\text{cross}}$ and $P_{\text{in}}$ denote the cross-polarized and input power, respectively. Lastly, $\Delta\lambda$ is the full width at half-maximum of the transmission minimum at the operating wavelength. Quantity | Label | Formula | Value
---|---|---|---
Transmittance | $T$ | $|E_{t}/E_{0}|^{2}$ | 0.001
Reflectance | $R$ | $|E_{r}/E_{0}|^{2}$ | 0.993
Absorptance | $A$ | $1-T-R$ | $\leq 0.006$
Quality factor | $Q$ | $\lambda/\Delta\lambda$ | 17.3
Field enhancement | $\Delta E$ | $\max(|E/E_{0}|)$ | 6.1
Numerical aperture | NA | $(k_{x}^{\text{max}}-k_{x}^{\text{min}})/k_{0}$ | $\leq 0.17$
Angular response | $\Theta$ | $\Delta\lambda/\lambda$ | 0.06
Polarization conversion efficiency | $\epsilon_{\text{pol}}$ | $100\%P_{\text{cross}}/P_{\text{in}}$ | 0.001%
Processing efficiency | $\eta$ | $|H_{xx}(k_{x}^{\text{max}},0)|$ | 79%
Switching contrast (amplitude) | $\delta$ | $H_{yy}(0,0)-H_{xx}(0,0)$ | 93%
Switching contrast (transmittance) | $\Delta$ | $T_{y}-T_{x}$ | 86%

###  S2.3 Isolated resonator

The optical response of an isolated silicon resonator of size given in the
main article was investigated by implementing the formalism explained in
section S1.2. The simulations were performed using the finite element method
within the wave optics module of COMSOL Multiphysics 6.1. The scattering
formulation was used to project the far-zone scattered electromagnetic field
onto the vector spherical harmonics on a virtual sphere of radius
$r_{0}=$750\text{\,}\mathrm{nm}$$ surrounding the resonator. Sommerfeld
boundary conditions and a perfectly matched layer were applied onto the
virtual sphere to remove unwanted reflections into the system and to simulate
an open domain. The finite meshing elements in the model had a a minimum size
of $5\text{\,}\mathrm{nm}$ and a maximum size of
$\lambda/$$8\text{\,}\mathrm{nm}$. The perfectly matched layer had a series of
5 meshing elements. The scattering cross section was obtained by integrating
the projections of the electromagnetic fields onto the vector spherical
harmonics and using Eqs. (S10)-(S15).

Figure S9: Optical response of an isolated silicon resonator in air. (a) The
3D radiation pattern. (b) The electric field strength (heat map) and direction
(arrows) when on-resonance in the $xz$-plane. (c) Same as b but for the
magnetic field. (d) The polar radiation plot in the $xz$-plane for $x$\- and
$y$-polarization. (e) A multipole decomposition of the scattering cross
section as a function of the wavelength. The results in (a)-(c) and (e) were
obtained with $x$-polarized illumination at the operating wavelength.

The results are given in Fig. S9 for the case of normally incident,
$x$-polarized plane waves illuminating the resonator at the operating
wavelength. The 3D (Fig. S9a) and polar radiation patterns (Fig. S9d) show
that majority of the light is concentrated in the forward direction. The
electromagnetic field behavior in the $xz$-plane shows field enhancement
within the resonator (Fig. S9b-c). The multipole decomposition of the
scattering cross section indicates that this is due to an electric dipole
resonance (Fig. S9e).

###  S2.4 Imaging simulations

Figure S10: Simulated tunable multi-modal microscopy with the metasurface.
(First row) The amplitude and phase masks of the input imaging target. (Second
row) The output images obtained with $x$-polarized illumination. (Final row)
The output images obtained with $y$-polarized illumination.

Simulations of tunable multi-modal microscopy were performed in MATLAB R2022a
by the formulations presented in the ‘Tunable multi-modal microscopy -
principles’ section of the main article and in the section S1.1. The optical
transfer function of the metasurface that was calculated in COMSOL
Multiphysics 6.1 (Fig. S7) was used to model the optical response of the
metasurface in Eq. (2). Three imaging targets were used in the simulations and
experiments. The first included an opaque amplitude mask with the word
‘Amplitude’ and a phase profile of the word ‘Phase’ (Fig. S10 \- top left),
which was presented in the main article. The second included the opaque word
‘Erythrocyte’ and the phase profile of a human red blood cell (Fig. S10 \- top
middle). Some of the experimental imaging results are presented in Fig. 1a of
the main article. The phase profile of the red blood cell was modelled using
the empirical optical properties provided in Ref. [9]. Lastly, the third
imaging target included the opaque word ‘T-Lymphocyte’ and the phase profile
of a leukaemic Jurkat cell (Fig. S10 \- top right) [10]. The phase profile of
the Jurkat cell was modelled using the empirical optical properties from Ref.
[11] and the imaging results for both simulation and experiment are presented
here.

The simulated imaging results are presented in Fig. S10. The images obtained
with $x$-polarized illumination contained both of the transparent and opaque
features of the input imaging target (Fig. S10 \- middle row). Each of the
phase profiles have a pseudo-3D appearance in the images. On the other hand,
only the opaque words are visible in the images obtained with $y$-polarized
illumination (Fig. S10 \- bottom row).

##  S3 Optical configurations

###  S3.1 Polarization

Figure S11: Polarization. Schematic of how the polarization is defined and
referred to in this article.

The metasurface operates under the excitation by linearly polarized
illumination. As in the main article, we define two coordinate systems. The
first is the global coordinate system with respect to the optical bench in the
experiment. The second is with respect to the metasurface defined by the long
axis of the silicon blocks, the short axis and the normal to the metasurface.
Assume that the illumination propagates along the global $z$-axis (optical
axis). The metasurface was placed in the global $xy$-plane with its long axis
aligned with the global $x$-axis. It was then tilted by
$2\text{\,}\mathrm{\SIUnitSymbolDegree}$ about the global $y$\- (short) axis.
Therefore, with this configuration, $x$-polarized illumination leads to the
production of phase contrast images, while $y$-polarized illumination leads to
bright field images. A schematic of the polarization is given in Fig. S11.

In the imaging experiments, the global $x$-axis was along the vertical
direction and the global $y$-axis was along the horizontal direction. The
metasurface was oriented with its long axis along the vertical direction (at
normal incidence), as depicted in Fig. S11. To change the polar angle of
incidence $\theta$, the metasurface was tilted about the horizontal direction.
The illumination was $x$-polarized when the input polarizer was along the
vertical direction. On the other hand, the illumination was $y$-polarized when
the input polarizer was horizontal. The effects of de-polarization and/or
cross-polarization by the spatial light modulator (SLM), imaging samples,
metasurface and optical components were measured to be negligible. For
example, the slight deviations on the polarization set by the polarizer due to
diffraction from the SLM was a weak effect.

###  S3.2 Spectrometry

Figure S12: Normal-incidence spectroscopy configuration. White light from a
halogen lamp was collimated by a microscope objective and linearly polarized.
The light transmitted by the metasurface was collected by a second microscope
objective and a lens. A second lens focused the transmitted light onto a
spectrometer. A beam-splitter was used to simultaneously relay an image of the
metasurface onto a camera by a lens.

The normal-incidence and angle-dependent spectroscopy experiments are detailed
in the Methods section of the main article. Here, schematics of their
experimental configurations are shown in Figs. S12 and S13.

Figure S13: Angle-dependent spectroscopy via back focal plane imaging. White
light from a halogen lamp was linearly polarized and illuminated the
metasurface through a microscope objective. The reflected light from the
objective back focal plane was imaged onto a spectrograph/CCD couple with a
tube lens and Bertrand lens pair. An analyzer was used before the spectrograph
to select specific components of the optical transfer function tensor.

###  S3.3 Imaging with a spatial light modulator

Figure S14: Experimental configuration for imaging involving a spatial light
modulator. The light from a super-continuum source was linearly polarized and
collimated by a microscope objective and two lenses. The lenses expanded the
beam to fill the aperture window of the SLM. An amplitude mask was placed in
front of the SLM. The light reflected by the SLM was focused into a microscope
objective by a lens. The microscope objective produced a small collimated beam
containing a de-magnified image of the SLM aperture. The metasurface was
placed in the image plane. A half wave-plate was placed before the lens to
change the polarization. A microscope objective and lens relayed a magnified
image onto a camera.

The experimental imaging results involving the SLM presented were obtained
with the configuration illustrated in Fig. S14. This schematic is also shown
in Fig. 3 of the main article.

###  S3.4 Biological microscopy

Figure S15: Bio-imaging configuration. The light from a super-continuum source
was linearly polarized and a half wave-plate was used to control the
polarization. The light illuminated a sample and the metasurface was placed
after the sample. A microscope objective and lens relayed a magnified image
onto a camera.

The experimental biological imaging results with the metasurface presented in
the main article were obtained with the configuration illustrated in Fig. S14.
This schematic is also shown in Fig. 4 of the main article.

##  S4 Additional experimental data

###  S4.1 Metasurface fabrication

Figure S16: Fabrication of the metasurface. (a) A photo of the metasurface
fabricated onto a coverslip. (b),(c) Scanning electron microscope images of
the metasurface. (d) A flowchart of the fabrication process. Plasma-enhanced
chemical vapor deposition is abbreviated to PECVD.

A flowchart of the fabrication process and pictures of the metasurface are
given in Fig. S16. The metasurface was fabricated on a
$10\text{\,}\mathrm{mm}$-thick glass substrate that was
$15\text{\,}\mathrm{mm}$ by $15\text{\,}\mathrm{mm}$ in size. A thin
$23\text{\,}\mathrm{nm}$ layer of indium tin oxide (ITO) was coated onto the
glass substrate to be used as a conducting protector for the electron beam
lithography process. Similarly, a thin $10\text{\,}\mathrm{nm}$ layer of
aluminium oxide (Al2O3) was coated onto the silicon to be used as a hard
protective mask for the etching process.

###  S4.2 Optical characterization

Figure S17: Angle-dependent spectroscopy via back focal plane imaging. (a) 2D
heat map of the reflectance counts from the metasurface as a function of the
wavelength and angle of incidence for $px$-polarization. (b) Same as a but for
$y$-polarization. The white dashed lines indicate the operating wavelength.

The angle-dependent spectroscopy experiments were performed with the back
focal plane imaging configuration depicted in Fig. S13 and detailed in the
Methods section of the main article. The metasurface was oriented such that
the angular range would be along the plane of incidence defined by the long
axis of the metasurface. The results (Fig. S17) show the reflectance counts as
a function of the wavelength and angle for input $x$\- and $y$-polarized
illumination. The former shows strong reflection at normal incidence at the
operating wavelength, which decreases in strength as the angle increases.
Meanwhile, the latter has a relatively low reflection. Plots of the line
profiles along the line segments in Fig. S17 are given in Fig. 2b-ii of the
main article in the form of transmission amplitude data.

###  S4.3 Performance metrics

Table 2: Experimental performance metrics of the metasurface. The experimental performance metrics obtained at the operating wavelength of $969\text{\,}\mathrm{nm}$. Quantity | Label | Formula | Value
---|---|---|---
Transmittance | $T$ | $|E_{t}/E_{0}|^{2}$ | $\geq 0.096$
Reflectance | $R$ | $|E_{r}/E_{0}|^{2}$ | $\leq 0.931$
Quality factor | $Q$ | $\lambda/\Delta\lambda$ | 19.29
Numerical aperture | NA | $(k_{x}^{\text{max}}-k_{x}^{\text{min}})/k_{0}$ | $\leq 0.17$
Angular response | $\Theta$ | $\Delta\lambda/\lambda$ | 0.05
Processing efficiency | $\eta$ | $|H_{xx}(k_{x}^{\text{max}},0)|$ | 78%
Switching contrast (amplitude) | $\delta$ | $H_{yy}(0,0)-H_{xx}(0,0)$ | 60%
Switching contrast (transmittance) | $\Delta$ | $T_{y}-T_{x}$ | 71%

The experimental performance of the metasurface was quantified through the
metrics presented in Table 2. These included the transmittance, reflectance,
quality factor, numerical aperture, angular response, processing efficiency
and the switching contrast. The results are tabulated in Table 2 and can be
compared to the simulations in Table 1.

###  S4.4 Imaging experiments

Figure S18: Tunable multi-modal microscopy experiments of samples produced by
a spatial light modulator - red blood cell. (a) The experimental images
obtained with the metasurface for different polarization states of the
illumination at the operating wavelength. (b) The input imaging target
included an amplitude mask and a phase profile on the spatial light modulator.
(c) The experimental image obtained in the absence of the metasurface. (d) The
image obtained by subtracting the image obtained with $x$-polarized
illumination from the image obtained with $y$-polarized illumination. Figure
S19: Tunable multi-modal microscopy experiments of samples produced by a
spatial light modulator - Jurkat cell. (a) The experimental images obtained
with the metasurface for different polarization states of the illumination at
the operating wavelength. (b) The input imaging target included an amplitude
mask and a phase profile on the spatial light modulator. (c) The experimental
image obtained in the absence of the metasurface. (d) The image obtained by
subtracting the image obtained with $x$-polarized illumination from the image
obtained with $y$-polarized illumination.

The multi-modal microscopy experiments involving the SLM were performed with
the three different imaging targets set out in section S2.4. The imaging
results with the ‘Amplitude’ and ‘Phase’ words in the imaging target were
presented in Fig. 3 of the main article. The imaging results of the red blood
cell are given in Fig. S18 and the results of the Jurkat cell are given in
Fig. S19.

The red blood cell (Fig. S18b) and Jurkat cell (Fig. S19b) were visualized in
the images obtained under $x$-polarized illumination (Figs. S18a and S19a).
The contrast of these features had a pseudo-3D appearance. On the other hand,
the words ‘Erythrocyte’ (Fig. S18b) and ‘T-Lymphocyte’ (Fig. S19b) were
visible in all of the images obtained in the experiment (Figs. S18a and S19a),
independent of the polarization. In particular, only these features were
visible in the images obtained under $y$-polarization. These images were
similar to the images taken in the absence of the metasurface (Figs. S18c and
S19c). The images were combined to qualitatively isolate the phase by
subtracting the images obtained with $x$-polarization from those obtained with
$y$-polarization (Figs. S18d and S19d).

###  S4.5 Biological imaging

As discussed in the main article, the thickness of the breast tissue sample
was a challenge for the imaging modalities used in the experiments. The
metasurface and the DIC microscope were capable of highlighting the
architectural structure within the tissue, even at the higher thickness
provided by the microtomy blade sectioning of the whole tissue blocks. This
capacity can allow for the identification of normal architectural patterns
within the tissue and those which are abnormal, e.g. tumours often form
aberrant architectural structural arrangements. However, identifying more
complex structures or lower order cellular detail is uncertain, as the imaged
area included only a small number of structures in a simple and benign
arrangement.

Instead, a more ideal application would be the preparation of cellular
material rather than formalin-fixed, paraffin embedded in-situ tissue that
requires a thicker tissue sample. In this case, the imaging samples would be
generated by smearing fresh tissue to directly transfer individual and
clusters of cells onto a glass slide, rather than an entire section of tissue.
This produces a thinner sample with cells juxtaposed on a background slide,
rather than remaining embedded in their native tissue environment. These
thinner samples would provide a greater differential between individual and
groups of cells and the background glass slide. Furthermore, it may facilitate
a better generation of cellular level imaging data.

###  S4.6 Experimental videos

Two experimental videos are included as supplementary material. The first
(Video S1) is a recording of an experiment made on the polarization switching
ability of the metasurface. By using the experimental configuration detailed
in the ‘Multi-modal microscopy - biological samples and polarization
switching’ sub-section of the Methods section of the main article, an image of
the metasurface was captured by a camera under normally incident illumination
at the operating wavelength. The video shows how the performance of the
metasurface changes as a function of the polarization.

The second (Video S2) video is a recording of an experiment made for imaging
C. elegans with the metasurface. The video shows a live nematode moving in the
sample, demonstrating the capacity for the metasurface to be used for dynamic
monitoring of samples.

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