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There are two main challenges in providing complete mapping of superoscillatory fields. First, since phase information can only be recovered in interferometric measurements, extreme stability of the interferometer is needed to obtain reliable data on fields with small spatial features and fast phase variations. Second, the superoscillatory fields are expected to have spatial features that are much smaller than the wavelength of light: a spatial resolution far better than that allowed by the Abbe–Rayleigh limit of a halfwavelength is required.
To meet these two challenges, we developed an original monolithic metasurface interferometry (MMI). In this technique, the superoscillatory field under investigation and reference wavefront needed for interferometry are created by the same planar metamaterial nanostructure, i.e., on the same monolithic platform, thus minimizing the issues of stability and alignment that are characteristic of freespace interferometers^{3335}.
Interference of the superoscillatory and reference fields creates a field distribution in free space that can be mapped with a linear detector array. The resolution of the detector array is determined by its pixel size and is insufficient for mapping. However, the field containing superoscillatory components and subwavelength features below the Abbe–Rayleigh diffraction limit is formed by the interference of freespace waves and can therefore be imaged with magnification by a lens with an NA exceeding that of the diffraction mask to collect all the wavevector components. We used a complementary metaloxide–semiconductor (CMOS) camera array with a pixel size of 6.5 µm together with a high magnification optics (×500), thus achieving a spatial resolution of 13 nm. It will be shown below that the monolithic metasurface interferometric technique allows for the detection of superoscillatory field features that are below 2% of the wavelength in size.
The key component of the monolithic metasurface interferometer is a metamaterial mask (Pancharatnam–Berry phase metasurface^{3638}) that simultaneously creates the tight superoscillatory focus and the reference wavefront for the interferometry (see Fig. 2a). It contains rows of identical scattering subwavelength slits oriented at either +45° or −45° with respect to the xaxis and has translation symmetry in the y dimension, thus working similarly to a cylindrical lens focusing light into a line focus. The working principle of the mask is shown in Fig. 2b. The mask is designed to be polarizationsensitive and only creates a superoscillatory field in crosspolarization to the incident wave. The field copolarized with the incident wave will propagate through the mask as a plane wave—with some attenuation. For example, when the metasurface is illuminated with xpolarized light ($E_x^{ {i}}$ on Fig. 2b), the phase of xpolarized transmitted light ($E_x^{ {t}}$) will be independent of the slit orientation: the xpolarized transmitted wave will remain a plane wave since the period is subwavelength and only the zero diffraction order is generated. However, the slits oriented at +45° and −45° to the xaxis will transmit ypolarized light with a π phase difference ($E_y^{ {t}}$ and $ E_y^{ {t}}$, see Fig. 2b). Therefore, the pattern of slits works as a binary phase grating for ypolarization. We define this arrangement as the TE configuration. Similarly, when illuminated with ypolarized light, the metasurface is a binary phase mask for the transmitted xpolarization (TM configuration).
Fig. 2 Monolithic metasurface interferometry with a superoscillatory Pancharatnam–Berry phase metasurface.
a Principle of monolithic metasurface interferometry: a single metasurface creates the field distribution under investigation and the reference wave in orthogonal polarization. These two wavefronts are made to interfere on a polarization sensitive detector. b Scanning electron microscopy image of the 40 µm by 40 µm metasurface fabricated in a 100 nmthickness gold film with focused ion beam milling (unit cell size 400 nm by 400 nm) that acts as a superoscillatory field generator and interferometric platform. Artificial colors indicate rows providing a binary 0/π phase shift in the transmitted light. The phase of light transmitted through the metasurface depends on the incident polarization and orientation of the slit, giving the opportunity to create a superoscillatory field for one of the transmitted polarizations and a plane wave for the orthogonal one, as explained in the text. A series of polarizationsensitive measurements of the intensity distribution of the diffracted light for different input polarizations allow for unambiguous recovery of the phase map of the superoscillatory fieldSuch a mask allows for a straightforward interferometry between the superoscillatory field and the reference plane wavefront that are mutually stable and inherently aligned by design. A 3D map of the intensity and phase can be recorded by measuring the intensity distributions at different distances from the mask using different input polarizations and polarizationsensitive detection.
Upon transmission through the metasurface, the xpolarized field suffers the same phase retardation regardless of the orientation of the slits and with the same intensity attenuation at all points due to the energy transfer into the crosspolarized field. Therefore, for the xpolarized field the metasurface is a homogeneous subwavelength grating of limited size (aperture). It will produce only a zeroorder diffraction field, which does not depend on the state of the polarization of light incident on the metasurface. Although the xpolarized field shows some variations from the plane wave due to aperture diffraction at the edges of the metasurface, it is a good reference field for interferometry as it has a phase close to that of a plane wave (see Supplementary Information section A for details) and a welldefined, easytomeasure intensity profile with no zeros.
In our experiment, the metasurface contains 100 rows of slits. The metasurface grating is designed with a particle swarm optimization algorithm to generate superoscillatory foci of the prescribed spot size, focal distance, field of view and depth of focus (see Materials and methods for the metasurface design details). Since the grating creates a superoscillatory focus for only one polarization while the transmitted light remains a plane wave for the other, the phase distributions in the TE and TM configurations $\varphi _{ {{TE, TM}}} = \arg \left({E_{y, x}} \right)$ can be mapped by measuring the intensity distribution $I\left({x, z} \right)$ of the interference pattern at a distance z from the mask for different polarizations of incident light (x, y, +45°, −45°, right and left circular) as follows:
$$ \varphi _{ {{TE}}} = {{atan}}\left( {\frac{{I_y^{ {{LCP}}}  I_y^{ {{RCP}}}}}{{I_y^{ + 45^\circ }  I_y^{  45^\circ }}}} \right) + k_0z $$ (1) $$ \varphi _{ {{TM}}} = {{atan}}\left( {\frac{{I_x^{ {{RCP}}}  I_x^{ {{LCP}}}}}{{I_x^{ + 45^\circ }  I_x^{  45^\circ }}}} \right) + k_0z $$ (2) Here, the second term on the righthand sides of Eqs. (2) and (3) comes from the reference plane wave, and the superscripts and subscripts correspondingly denote polarization of the incident light and detection light. See Supplementary Information section B for the operating principle of the phase retrieval.
The results of mapping the interference patterns $I\left({x, z} \right)$ for the TE configuration can be found in Fig. 3, which shows performance of the metasurface evaluated by a finitedifference timedomain (FDTD) calculation. When the incident light at a wavelength of 800 nm is xpolarized, the ypolarized component of the diffracted wave contains a superoscillatory hotspot 10 µm away from the metasurface. The focal spot has a fullwidth at halfmaximum (FWHM) of 0.42λ, which is well below the Abbe–Rayleigh diffraction limit λ/2NA = 0.56λ for a cylindrical lens with a numerical aperture corresponding to the experimental situation of NA = 0.89 (20 µm wide lens with a focal distance of 10 µm). Under ypolarized illumination, the ypolarized component of the diffracted wave is the reference field that we use for interferometry. For an infinitely long grating, it would show no structural features, while the minor variations in the transmission amplitude observed experimentally are due to the aperture effects. Its phase is uniform and therefore intensity variations do not affect the accuracy of the phase retrieval (see Supplementary Information section A). With circularly polarized incident waves, the diffraction patterns $I_y^{ {{RCP, LCP}}}\left({x, z} \right)$ originate from the interference of the superoscillatory field and the reference wave with an initial phase difference of ±π/2 between them, depending on the handedness of the incident polarization. Similarly, for the incident linear polarization at ±45°, when we measure $I_{y}^{\pm 45^{\circ}}\left({x, z} \right)$, the phase difference between the superoscillatory and reference fields becomes 0 and π, respectively. Now the phase map φ_{TE}(x, z) of the superoscillatory field can be recovered from $I_y^{ {{RCP, LCP}}}\left({x, z} \right)$ and $I_{y}^{\pm 45^{\circ}}\left({x, z} \right)$ maps using formula (2).
Fig. 3 Superoscillatory field generated by the metasurface under different illumination conditions.
a FDTD simulation and b experimental data. The figures show the x–z crosssection of the ycomponent intensity distribution in the interference pattern I_{y}(x, z) for different incident polarizations (indicated in the corners of the maps—TE configuration). Note that the formation of the superoscillatory hotspot (highlighted in the blue box) located at a distance of 10 µm from the metasurface is best manifested for orthogonal incident polarization (E_{x}). The quasiuniform field map for the parallel incident polarization (E_{y}) is multiplied by a factor of 4 and behaves like a reference plane wave for creating the interferogram with the signal superoscillatory field. The FDTD is computed and the experimentally measured maps show good agreementIn our experiment, we used an 800 nm wavelength diode laser as an optical source and mapped the interference pattern I(x, z) with a CMOS camera placed on a nanometric translation stage and equipped it with a ×500 magnification optical system. See Supplementary Information section C and Materials and methods for the detailed optical characterization setup. The corresponding experimental results for the intensity maps are given in Fig. 3b. Good agreement with the calculated maps is found in all field patterns. The copolarization light shows intensity variations due to aperture diffraction on the mask edges. Some asymmetry in the pattern in Fig. 3b is due to imperfections in the incident wavefront. From a quantitative calculation, the mean square differences between a plane wave wavefront and the simulated and measured field maps were found to be 11.7% and 24.1%, respectively. Here, the superoscillatory hotspot (upper left panel in Fig. 3b) is also observed at z = 10 µm and its FWHM is measured to be 0.43λ, which is only about a 2% difference from the computed size of the hotspot.

Figure 4a shows the computed and measured x–z crosssections of the intensity map in the near vicinity of the superoscillatory focus, annotated by the blue box in Fig. 3. Here, we can see the first characteristic feature of the superoscillatory optical field, namely, the high localization of the field. The hotspot size in the xsection is smaller than that allowed by the Abbe–Rayleigh limit (because it does not take superoscillation into account). The focus is surrounded by fringes, similar to how the focal spot of a conventional lens of finite size is surrounded by the oscillating Airy pattern. However, here the fringes are more densely spaced than in the Airy pattern and much more extensive fringes are present. Indeed, superoscillatory hotspots are always surrounded by intense halos or fringes^{15}. At the focal plane, the intensity of the first sidelobe is 17.6% (simulation) and 16% (experiment) of the peak intensity of the central hotspot.
Fig. 4 Four characteristic features of a superoscillatory field. (Top row) simulation; (Bottom row) experiment.
a High localization of the field can be seen from the intensity map E_{y}^{2} depicting focus with a FWHM of 0.42λ, smaller than that allowed by the Abbe–Rayleigh limit. The vertical dashed line indicates the plane of the focus. b Phase singularities with topological charge of m = ±1 (green dots) are seen on the phase maps arg(E_{y}) in the lowintensity areas of the superoscillatory field. c Gigantic local wavevectors k_{x}/k_{0} at the focal plane are calculated from the phase gradient in xdirection. Superoscillatory values of local wavevectors are highly localized in zones of order λ/100. d The energy backflow (retropropagation) areas are painted white. They are substantially subwavelength in size (~λ/20 along xdirection) and correspond to negative values of k_{z} calculated from the phase gradient in the zdirection. Insets with the black background show a zoomedin view of the timeaveraged Poynting vectors near the phase singularity, and the backflow regions are shaded in blue. Dashed green lines indicate tangent to the retropropagation areas at the point of their intersections with phase singularitiesThe phase φ_{TE}(x, z) of the electromagnetic field is rapidly changing near the superoscillatory focus, as shown in Fig. 4b. Here, one can observe a close match between the computed and experimentally measured phase maps that were retrieved from the intensity maps in Fig. 3b using formula (2). On the phase maps, one can clearly observe the second characteristic feature of superoscillatory optical fields: they are accompanied by phase singularities. At the lowintensity regions near the focus, one can see four phase singularities identified by green circles (Fig. 4b). When moving along the loop encircling the phase singular points, the phase changes by 2π. The two singularities in the upper part of the phase map have a topological charge of m = +1, while in the lower part they have topological charge of m = −1.
As the third characteristic feature of a superoscillatory optical field, we observed gigantic local wavevectors in the field maps with values far exceeding k_{0} = ω/c. First, we calculated the transverse wavevector k_{x} as the xcomponent of the gradient of the computed and measured the phase values. In the experiment, the pixilation of the phase mapping was 13 nm along the xdirection (determined by the effective pixel resolution of the detector) and 10 nm along the zdirection (scanning step size of the piezo stage). The normalized transverse local wavevector k_{x}/k_{0} at z = 10 µm is given in Fig. 4c after data smoothing and interpolation with a step size of 5 nm. From there we see that the k_{x} near the phase singularity is more than an order of magnitude higher than k_{0}. Here, the computed and experimental data are in good agreement qualitatively. Although the data are somewhat smaller in the experiment, very large wavenumbers beyond the spectrum are still observed. Also note that the phase retrieval and wavevector mapping are robust to noise, see Supplementary Information section D on the noise sensitivity analysis.
The presence of the fourth characteristic feature of a superoscillatory optical field: the existence of energy backflow (retropropagation) areas near the superoscillatory focus, which can be derived from the mapping of longitudinal wavevector k_{z}. On the x–z maps we painted the areas of energy backflow as white zones in Fig. 4d. Indeed, we observed that k_{z} can have negative values near the phase singularities. Since the Poynting vector is parallel to the local wavevector in freespace, negative values of k_{z} mean energy backflow (see insets presenting the computed time averaged Poynting vector maps $\left\langle S \right\rangle = {\Re} \left({E \times H^ \ast /2} \right)$ near phase singularities). As can be clearly seen in the inset of Fig. 4d, phase singularities are pinned to the energy backflow regions, as predicted in ref. ^{32}. Here, one can see that the incident energy flow is "trapped" and circulates without propagating in the forward direction (compare with energy flow near the plasmonic nanostructure, see Fig. 1). As predicted by Berry for a general case of interfering multiple waves^{32}, "The boundaries of the retropropagating regions include the phase singularities … and are tangent to the z direction at these points" (see green dashed lines on Fig. 4d. They are consistently directed along z, which confirms the accuracy of the experiment). Here, we also confirm another powerful observation from the same paper that "the regions of backflow are considerably smaller than the wavelength; this reflects the wellknown fact that in the neighborhood of phase singularities wavefunctions can vary on subwavelength scales." Indeed, backflow areas are only about λ/20 in size along the xdirection but are still satisfactorily resolved and their positions are accurately mapped on the computed locations. It is also noteworthy that due to the translation symmetry no energy backflow is observed in the plane perpendicular to the x–z plane. Note that similar optical vortices and energy backflow phenomena also exist in the longitudinally polarized field^{39}.
The existence of phase singularities and energy backflow zones pinned to optical superoscillations gives a qualitative insight into the mechanism of focusing beyond the Abbe–Rayleigh limit. Two singularities that are close to the superoscillatory focus are located in the areas of diminishing intensity that define the boundary of the focus. At the superoscillatory focus, the backflow depletes the area where flow propagates in the forward direction, thus narrowing the focus beyond the conventional diffraction limit.
Similar results for the TM configuration are presented in Supplementary Information section E, where all the four characteristic features of superoscillatory fields, including the high localization of field, phase singularities, gigantic local wavevectors and energy backflow, are also experimentally observed. In addition, all these features can also be observed near resonant plasmonic nanoparticles. See an example in Supplementary Information section F.
Nanoscale interferometry of superoscillatory fields
Four features of superoscillatory fields

The superoscillatory Pancharatnam–Berry phase metasurface is designed using the binary particle swarm optimization (BPSO) algorithm. The twodimensional metasurface at the x–y plane is divided into N = 50 pairs of equally spaced rows of slits (Δx = 400 nm) placed with mirror symmetry with respect to the x = 0 plane. Each row has slits oriented either at +45° or −45° with respect to the xaxis and provides phase delays of π and 0, respectively. The BPSO algorithm optimizes the light field distribution created by the metasurface near the focal plane when illuminated with a plane wave. The merit function to be optimized is defined as
$$ I^{ {{tar}}}\left( {x, z} \right) = \left[ {{{sinc}}\left( {ax} \right)} \right]^2{{exp}}\left[ {  \left( {z  z_{ {f}}} \right)^2/b^2} \right] $$ (3) where z_{f} is the desired focal length, a = 0.886/FWHM, $b = D{{/2}}\sqrt {\ln 2}$, FWHM is the fullwidth at halfmaximum of the hotspot size and D is the depth of focus.

The designed metasurface was fabricated by focused ion beam milling (FIB, Helios 650, Hillsboro, OR, USA) on a 100 nmthick gold film deposited on a glass substrate using thermal evaporation (Oerlikon Univex 250, Cologne, Germany) with a deposition rate of 0.2 Å/s. The FIB writing voltage is 30 kV and the current is 7.7 pA.

The laser source is a diode laser with an emission wavelength of 800 nm (Toptica DLC DL pro 780, TOPTICA Photonics AG, Munich, Germany) and a linewidth of 100 kHz. A linear polarizer (P1) and a quarter waveplate (QWP) or half waveplate (HWP) are used to produce the desired incident polarization. The field distribution created by the metasurface was mapped with a highresolution camera (Andor Neo sCMOS, Andor Technology Ltd., Belfast, UK; 2560 × 2160, pixel size 6.5 µm) and a highmagnification apochromatic planar objective corrected for field curvature (Nikon CFI LU Plan APO EPI (Nikon Instruments Inc., Melville, NY, USA) ×150, NA = 0.95) with a ×4 magnifier making distortion of the images practically negligible. The actual magnification factor is calibrated to be ~500, corresponding to an effective pixel size of 13 nm. Another polarizer (P2) is inserted into the optical path before the camera to select the desired detection polarization state (E_{y} in the TE case, E_{x} in the TM case). The field maps were obtained by zscanning with a step size down to 10 nm using a piezo stage (PI E517, Physik Instrumente GmbH & Co., Karlsruhe, Germany).