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# Engineering phase and polarization singularity sheets

## Abstract

Optical phase singularities are zeros of a scalar light field. The most systematically studied class of singular fields is vortices: beams with helical wavefronts and a linear (1D) singularity along the optical axis. Beyond these common and stable 1D topologies, we show that a broader family of zero-dimensional (point) and two-dimensional (sheet) singularities can be engineered. We realize sheet singularities by maximizing the field phase gradient at the desired positions. These sheets, owning to their precise alignment requirements, would otherwise only be observed in rare scenarios with high symmetry. Furthermore, by applying an analogous procedure to the full vectorial electric field, we can engineer paraxial transverse polarization singularity sheets. As validation, we experimentally realize phase and polarization singularity sheets with heart-shaped cross-sections using metasurfaces. Singularity engineering of the dark enables new degrees of freedom for light-matter interaction and can inspire similar field topologies beyond optics, from electron beams to acoustics.

## Introduction

Optical phase singularities are zeros in a complex scalar field—regions of darkness surrounded by light1. These singularities are ubiquitous in complex wave systems2 and typically arise due to rapid or discontinuous phase variation. One common manifestation of this is encountered in helical beams with orbital angular momentum (OAM), which exhibit a linear singularity along the optical axis, around which the wavefront (and thus the Poynting vector) swirls and forms an optical vortex. Phase structures of this nature have attracted much attention owing to the non-intuitive behavior they can imprint upon the surrounding field3. For instance, due to ring singularities around the Airy disk in the focal plane of a lens, there are locations near the singularity which have time-averaged Poynting vectors that point back towards the lens, thereby exhibiting locally backward energy flow4,5. Furthermore, superoscillations, whereby a bandlimited signal exhibits rapid spatial variation that can be arbitrarily larger than its maximum Fourier component6, also typically occur by virtue of nearby singularities7. Moreover, phase singularities exhibit different topologies as compared to bright regions and obey different constraints. For example, the diffraction limit constrains the focusing of light. There is no diffraction limit for dark: one can make phase singularities arbitrarily localized and measure their locations with deeply subwavelength precision, limited only by the signal-to-noise ratio of the measuring apparatus8. Indeed, measuring an intensity minimum is far more precise than measuring the displacements of finite-width beams of light9, an insight that enables a host of applications. Molecular scale imaging enabled via the MINFLUX technique10 can infer the position of a fluorescent particle from the spot of minimum emission when the particle is illuminated with a doughnut-shaped pump beam with zero on-axis intensity. Recently, the precise position of the singularities was also exploited to create a subwavelength ruler8 that can resolve displacements to better than λ/800.

Optical phase singularities such as those along the axis of OAM beams are widely studied one-dimensional (1D) singularity lines. Much is also known about the stochastic properties of 1D singularities in random fields such as laser speckle11. However, not all phase singularities are 1D. One-dimensional structures are indeed the most common singular features, as they are stable under small perturbations of the scalar field12. Although less common, point (0D) and surface (two-dimensional (2D)) singular structures also exist. Similar to high-order optical vortices, these rare structures are structurally unstable as a small-field perturbation can deform its singular region, removing the singularity altogether, or reducing it to a collection of stable 1D singularities. These singular structures are typically only observed in systems with high symmetry (“degenerate” cases), such as the planar destructive interference fringes in Young’s double slit experiment, as pointed out by Nye and Berry1 in their seminal paper predating singular optics, and the radial nodes in cylindrically symmetric diffraction-less Bessel beams13.

Here we show on-demand engineering of sheet (2D) singularities in scalar fields without the symmetry constraints. We do so by enforcing large, directed phase gradients at the target singular locations to produce dark sheets. We perform this phase gradient maximization in linearly polarized fields to produce phase singularities and separately in the paraxial vectorial electric field to produce transverse polarization singularities. We validate these numerical predictions experimentally using metasurfaces which implement the required wavefront profile. In contrast to existing studies, our approach does not specify the analytical mathematical description of the field in the vicinity of the singularities14,15, as in the construction of a superoscillating function to form an array of optical vortices16. Instead, we sculpt singularities based on phase gradients at the position of the singularity itself. This enables the dark regions to be realized with remarkably high contrast and fidelity. Singularity engineering points to the possibility of designing complex beams combining structured light and structured dark, and may inspire exotic field topologies in wave physics beyond optics, from electron beams to acoustics.

## Results

### Singularity geometries and structural stability

Singularities refer to undefined parameter(s) of a field. Phase singularities occur where real and imaginary parts of a scalar time-harmonic (with e−iωt convention here) field E(r) = Re[E(r)] + iIm[E(r)] are simultaneously zero, thus leaving the phase ϕ(r) = arg(E(r)) undefined there. In most complex three-dimensional (3D) scalar fields, the points at which the real (or imaginary) part of the field vanishes forms a 2D surface: a zero-isosurface. The optical singularities thus occur at the intersection of these two zero-isosurfaces (Re[E(r)] = 0 and Im[E(r)] = 0). The intersection may occur at exactly one point where the zero-isosurfaces touch, forming a point singularity (Fig. 1a). By bringing the two zero-isosurfaces closer together, the intersection locus becomes a closed loop (Fig. 1b). Linear singularities can also be open, extending to infinity, as in the widely known fundamental Laguerre–Gaussian0,1 (LG0,1) vortex beam (Fig. 1c), where the real and imaginary zero-isosurfaces only cross on the optical axis. Cross-sections of these 1D singularities demonstrate that the complex phase swirls in either a clockwise or anticlockwise manner around the singularity locus, accumulating ±π/2 over each of the four quadrants defined by the zero-isolines. Higher-order versions of these 1D geometries, such as the second-order LG0,2 beam, can be produced if the intersection line coincides with more than one sheet from each of the real and imaginary zero-isosurfaces (Fig. 1d). Notably, when the two zero-isosurfaces coincide, a 2D sheet singularity is produced (Fig. 1e).

First-order 1D singular structures differ from the other topologies in terms of their structural stability. Consider adding a small complex number to each of the scalar fields in Fig. 1. Physically, it can be considered an additional plane wave in the long wavelength limit, which, e.g., may arise from a defect or scatterer in the medium. The addition displaces and distorts the zero-isosurfaces. For 0D and 2D topologies, this shift misaligns the precise orientation of the zero-surfaces, destroying the singularity altogether (when the zero-surfaces do not intersect) or producing 1D singular lines (where the new isosurfaces intersect). On the contrary, the perturbation to the 1D singularity fields with first-order singularities in Fig. 1b or c merely causes the intersection to be displaced in space, preserving the existence of the 1D singularity. Higher-order 1D singularities (as in Fig. 1d) also dissociate into collections of first-order 1D singularities (Fig. 1c) in realistic circumstances under perturbation17. In summary, first-order 1D singularities are robust against field perturbations, whereas 0D, higher-order 1D, and 2D singularities are not.

One-dimensional singularities are commonly called topological singularities because they possess an invariant quantity, the topological charge, which is conserved under small-field perturbations and with propagation. For simplicity, we focus here on paraxial fields with a well-defined transverse plane. The topological charge s = C(ϕ/2π)·dr is computed around an enclosing loop C (usually chosen to lie on the transverse plane and defined to be positively oriented in the clockwise direction) so that the singularity line penetrates the interior of C. When there are no singularities along C, ϕ is continuous on the closed loop and must return to its original value modulus 2π. Thus, s is an integer and can be positive, negative, or zero. This same concept can be extended to 2D singularities by picking C large enough to encircle the 2D structure on the transverse plane. Care must be taken, however, when dealing with 2D singularities that have a closed transverse cross-section, as for calculating s, C must be chosen to have both an inner and outer loop (Supplementary Fig. 1). s is conserved under small field perturbations and with wave propagation, provided no singular structures cross C, and is equal to the sum of the topological charges of the singular structures penetrating the interior of C.

Although optical devices can be designed to shape lighted regions of a field according to one’s specifications through iterative techniques such as computer-generated holography, the Gerchberg–Saxton (GS) algorithm18,19 or adjoint optimization20, or through non-iterative means such as in freestyle laser traps21,22, sculpting the dark is more challenging. Iterative light-shaping numerical techniques rely on the reverse propagation of a desired light-field distribution to provide information on how to improve the design of an optical device in the next iteration. These techniques work well if the desired output optical intensity profile comprises lighted regions but fail if the desired pattern is dark or associated with little optical intensity; if there are singular regions in the desired optical output, reverse propagation of the output from these regions will result in zero information provided to the iterative algorithm. One might argue that these limitations can be bypassed by specifying the way the nearby fields are distributed so that the intensity goes to zero at the desired singularity locations. By doing so, one can then construct the closest wave-equation solution to the predefined phase and amplitude distribution using analytic techniques23. However, in specifying one spatial pattern adjacent to the singularity, one will reduce the space of acceptable designs by excluding other optical fields that also contain the desired singularity structure, potentially excluding better-behaved fields which more closely approximate the desired field distribution. In short, one must go beyond the standard techniques of optical engineering to manipulate the dark side of light.

### Singularity shaping strategy

Calculating fields with large phase gradients also requires special care in taking the derivative. Finite difference methods should not be used to estimate the phase gradient near singularities where the phase gradients are diverging because it is susceptible to errors, in particular due to the 2π phase periodicity. In our approach, we perform all calculations on an algorithmic differentiation (backpropagation) platform. We record the mathematical operations performed in connecting variables (e.g., device design parameters and their positions in space) to their results (e.g., complex field values or objective function values). We then traverse this record using the chain rule for derivatives to obtain the exact numerical derivative of any calculated value with respect to any other variable24.

It is important to mention that the 2D singularities we study here may not be absolute mathematical singularities in the sense that the field value along the singularity is small but may not be identically zero. This can be reconciled as follows: first, we chose to explore the parameter space of physically realizable light fields generated through a finite aperture and to find the closest field solutions that maximize the phase gradient locally on a finite set of points. Mathematically exact sheet singularities that are infinitely extended over space typically require an infinite aperture to be replicated perfectly, such as those associated with 1D diffraction patterns1 or those similar to Bessel beam nodes13 and LG radial nodes25. Second, although the phase gradient can become arbitrarily large during optimization, we truncate the optimization process early to avoid dealing with infinities and numerical precision limitations. These approximate singularities may not have zero field value but are close enough to zero with large phase gradients for physically interesting realizations. It is noteworthy that this is also the case for 3D volumetric singularities: although volumetric singularities are not mathematically possible (see Supplementary Note 2), “perfect” optical vortices (i.e., vortices with dark core radii that are both independent of the topological charge and where the largest field gradients are located) with a core volume containing very low light intensity have been designed and generated experimentally26.

Although not the focus of this study, we note that one way to deterministically position 0D point singularities is to place them on the axis of a cylindrically symmetric field. Both amplitude minimization and phase gradient maximization can be applied to produce these singularities. Supplementary Fig. 2 plots a simulated cylindrically symmetric field with three deterministically placed 0D singularities. More details are available in the Supplementary Methods.

### Engineered phase-singularity sheet

Similar to asymmetric optical vortices with fractional topological charge or high-order vortices with topological charge >1, the 2D heart-shaped singularity sheet is unstable with propagation28,29 and hence has a finite length in the axial direction. This instability with propagation is not a general characteristic of 2D sheet singularities; Bessel and LG beam nodes and 1D diffraction patterns are notably stable with propagation. The topological charge of our heart singularity sheet measured at the target plane is s = 0, computed using the inner and outer curves in Supplementary Fig. 7, indicating that equal numbers of vortices with s = +1 and −1 are formed following the break-up of the sheet singularity. The physical interpretation of this quantity is that it is an estimate of the mean OAM per photon l associated with a region comprising the singularity, in units of ħ. We can explicitly compute l over the cross-sectional area A between a set of inner and outer curves (Supplementary Fig. 7) using the method proposed by Allen and Padgett30, where l is given by the ratio of the time-averaged OAM, Jz, to the time-averaged energy per unit length, W (Eqs. (13)).

$${J}_{z}={\epsilon_0}\iint_{\!\!\!\!{\rm{A}}}dxdy({\bf{r}}\;\times {\rm{Re}}[{\bf{E}}\;\times\;{\bf{B}}^\ast ]/2)\cdot \hat{z}$$
(1)
$$W={\rm{c}}{\epsilon_0}{\iint }_{\!\!\!\!{\rm{A}}}dxdy({\rm{Re}}[{\bf{E}}\;\times\;{\bf{B}}^\ast ]/2)\cdot \hat{z}$$
(2)
$$l={\rm{\omega }}{J}_{z}/W$$
(3)

We obtain l = −0.0011, which is very close to the topological charge of s = 0 obtained by line integration.

### Experimental realization of phase-singularity sheet

We validate the numerical predictions experimentally by fabricating a transmissive metasurface to realize the required phase profile in Fig. 3b. Metasurfaces are not the only means to produce sheet singularities; any wavefront shaping device, which can sample the highest spatial frequency of the target pattern, can also be deployed. The metasurface comprises cylindrical nanopillars of TiO2 on an SiO2 substrate. Scanning electron micrographs of the full metasurface and the nanopillar arrays are exhibited in Fig. 4a, and the experimental setup used to profile the singularity sheet produced is displayed in Fig. 4b. The intensity and reconstructed phase profiles produced by the metasurface at the z = 10 mm plane are plotted in Fig. 3e, h, respectively. The phase profile was obtained through single-beam multiple-intensity reconstruction31 applied to the measured 3D intensity map, as depicted in Fig. 4c. The intensity and phase profiles show close correspondence to those obtained through simulation in Fig. 3d, g. Furthermore, the rapid intensity decay along the singularity sheet (~ −35 dB), stemming from the phase gradient maximization, can be observed in the plots. Notably, the continuity and uniformity of the dark contour is difficult to achieve using conventional computer-generated holography techniques that rely on amplitude minimization. This can be seen by comparison to Fig. 3f, i, which exhibits the best field intensity and phase obtained by using the GS phase retrieval algorithm to design a similar heart-shaped singular trajectory just on one transverse plane. Supplementary Fig. 8 demonstrates how the GS algorithm is unable to replicate fine intensity features over the transverse plane due to the limited range of transverse spatial frequencies and further details on this can be found in Supplementary Note 3. In addition, Supplementary Fig. 9 displays the cross-sectional intensity and phase profiles of the singularity at various z-positions alongside the numerical predictions and are in very good agreement.

### Engineered polarization singularity sheet

The process for engineering 2D phase-singularity sheets is also directly applicable to engineering 2D polarization singularity sheets. A detailed review of the connection between phase and polarization singularities is provided by Ruchi et al.32. Polarization singularities can take multiple forms, but here we focus on C-point singularities in which the polarization azimuth Ψ (in a paraxial field, with negligible Ez contribution) is singular12,33. Ψ is the angle that the major axis of the polarization ellipse makes with the transverse x-direction (Supplementary Fig. 10). Together with the ellipticity angle θ, which determines the eccentricity and handedness of the ellipse, the pair (Ψ, θ) parameterizes the space of fully polarized light. To deploy phase gradient maximization, we need to identify a complex scalar field σ such that arg(σ) Ψ. This complex field will exhibit the same geometries in Fig. 1. Maximizing the phase gradient of σ then maximizes the azimuth gradient and produces polarization singularities. One such field is σ = |Ex | 2 − |Ey | 2 + 2i·Re(ExEy*) for which arg(σ) = . This can be written as σ = s1 + is2 using the Stokes polarization parameters, which comprise four experimentally measurable intensities that parametrize the full space of polarized (and partially polarized) light. They are defined in Eqs. (4)–(7) using the left-handed convention34.

$${s}_{{0}} \!\!= \!\!{I}_{x}+{I}_{y}=|{E}_{x}{|}^{2}+|{E}_{y}{|}^{2}$$
(4)
$${s}_{{1}} \!\!= \!\!{I}_{x}\mbox{-}{I}_{y}=|{E}_{x}{|}^{2}\mbox{-}|{E}_{y}{|}^{2}$$
(5)
$${s}_{{2}} \!\!= \!\!{I}_{{45}{^\circ }}\mbox{-}{I}_{{-}{45}{^\circ }}=2\cdot {\rm{Re}}({E}_{x}{{E}_{y}}^{\ast })$$
(6)
$${s}_{{3}} \!\!= \!\!{I}_{{\rm{LCP}}}\mbox{-}{I}_{{\rm{RCP}}}=2\cdot {\rm{Im}}({E}_{x}{{E}_{y}}^{\ast })$$
(7)

where Ij refers to the intensity of the polarization component projected in the j direction. For fully polarized light, s02 = s12 + s22 + s32, which allows all polarization states to be assigned a unique point on a sphere in (s1,s2,s3) space—the Poincaré sphere (Supplementary Fig. 10). Lines of constant Ψ are longitudes on the sphere. Just as globe longitudes intersect at the poles so that the longitudinal coordinate is undefined there, Ψ is singular at the Poincaré sphere poles at which circularly polarized light resides. There, s1 and s2 go to zero, in analogy to the vanishing of the real and imaginary parts of a scalar complex field at a phase singularity. It is noteworthy, however, that the overall field intensity at the polarization singularity can be non-zero, because s3 may not vanish.

Figure 5 plots transverse cross-sections of an engineered polarization singularity sheet with a jump in Ψ tracing the shape of a heart (real and imaginary zero-isosurfaces of the σ-field are plotted in Supplementary Fig. 3). The cross-sectional heart-shape is identical to that of the phase-singularity sheet in Fig. 3, is also centered at z = 10 mm, and is designed to be generated by a plane wave at λ0 = 532 nm and with 45° linear polarization incident on a patterned 0.4 mm × 0.4 mm square aperture (at z = 0 mm) that serves as a spatially variant waveplate. The paraxial approximation (qualitatively supported by the target plane distance being much larger than the aperture size) is justified, as the energy density associated with the longitudinal Ez in the volume of interest is much smaller (in this case, <0.05%) than that of the transverse Ex and Ey components. The polarization singularity sheet is inversely designed by tuning the properties (phase delay, fast axis angle) of the spatially variant waveplate on a pixel-by-pixel level. The optimized parameters are plotted in Supplementary Fig. 11. Figure 5 plots Ψ, θ, and the field intensity (s0) of the simulated singularity sheet at and around z = 10 mm in the “Sim.” columns. The Ψ jump across the singularity has a magnitude of π/2 (as opposed to π, as in the phase-singularity sheet), because a sign flip in σ means that the change in arg(σ) = 2Ψ is π. The polarization singularity can also be visualized by examining the transverse gradient of the azimuth |Ψ| (Supplementary Fig. 4), which displays rapid spatial variations far exceeding k0.

### Experimental realization of polarization singularity sheet

As designed, the polarization singularity requires a device that can control the polarization variation of an optical wavefront, point-by-point. Although a spatial light modulator can perform this by structuring and then superimposing two coherent beams with orthogonal polarizations, a single spatially variant waveplate is best suited for this application. One realization of a spatially variant waveplate is a polarization-sensitive metasurface. Transparent birefringent nanofins on a metasurface behave locally as waveplates that perform a unitary transformation of the incident electric fields35. We fabricated this metasurface using TiO2 nanofins on SiO2 (micrographs in Fig. 6a). The polarization singularity structures generated by the metasurface are analyzed at 41 z-positions around z = 10 mm using rotating quarterwave plate polarimetry36 in the experimental setup shown in Fig. 6b. The measurements of Ψ, θ, and intensity (s0) at and around z = 10 mm are plotted in Fig. 5 in the “Exp.” columns and show good agreement to the simulated patterns. The Stokes’ parameters comparison between simulations and experiment are plotted in Supplementary Fig. 12.

## Discussion

Here we provide experimental demonstrations of on-demand singularity shape engineering beyond simple curved or straight lines. We have achieved 2D singularity sheets with precisely engineered dark intensity profiles in both scalar and vector fields by maximizing phase gradients orthogonal to the desired sheet structures. On-demand singularity engineering opens up a vast set of possibilities in wide-ranging fields: the sensitivity of singularity sheets to perturbations and inhomogeneities in the propagation medium can be exploited to reconstruct density fluctuations and currents in transparent or weakly scattering media. This can be performed by sending a precisely engineered singularity structure into a weakly scattering medium, and then capturing the distorted light field with a phase-sensitive technique. The singularities may also be strategically introduced in holography to improve the light/dark contrast, or in microscopy and telescopy to selectively suppress signals from parts of the image, as an extension of the vortex coronagraph from astronomy37,38. Such shaped singularities may also be useful in conjunction with MINFLUX-like techniques to localize elongated fluorescent emitters (e.g., quantum rods), both in space and in rotation angles with deeply subwavelength precision. By rotating and translating a pump beam with an extended singularity feature around an elongated emitter, then finding the position and orientation that minimizes the fluorescent signal, one can identify the associated position and orientation of that elongated emitter with wavelength-independent precision. Singularity engineering may also be deployed in shaping radiofrequency or acoustic emission patterns so as to deterministically produce and manipulate dead or quiet zones.

In particular, singularity sheets may find application in atomic trapping of neutral atoms using intensity gradients. The vast majority of optical dipole traps for cold atoms are “red” traps, which trap atoms in arrays of tightly focused spots of light, where the light is red detuned from the dipole resonance. Blue traps with 3D spatial confinement, which trap the atoms in a dark spot surrounded by blue-detuned light, are much harder to realize and techniques using a single structured beam have only been able to produce single blue traps thus far39,40,41. Deterministic singularity engineering of singularities produced by a single incident beam and metasurface may prove to be useful in generating blue trap arrays or exotic shapes on demand.

Beyond optics, singularities are found in many complex wave systems, such as acoustic, particle beam, fluidic, and plasmonic systems. Similar strategies of optimizing spatial gradients may enable one to engineer complex fields imbued with the strange behavior of nearby singularities.

## Methods

Detailed methods for device optimization and fabrication are found in the Supplementary Methods.

### Simulations

Full vectorial field profiles were obtained by discretization of the vector diffraction propagation integrals27 on the Tensorflow automatic differentiation platform42. The free parameters for each optimization are either the phases at each pixel on a phase mask or the waveplate properties (phase shift in the fast and slow axes, rotation angle of the fast axis) at each pixel on a spatially varying waveplate. The optimization objective function to be maximized is a smooth approximation to a minimum function applied to the list of directional phase derivatives at specified points in the field. Optimization is performed using gradient descent through the Broyden–Fletcher–Goldfarb–Shanno algorithm43 with iterative refinement applied to the parameter distribution, to reduce the risk of converging to a poorly performing local optimum.

### Metasurface fabrication

The metasurfaces are fabricated out of TiO2 on a glass substrate44,45. The glass substrate is a 0.5 mm-thick JGS1-fused silica. The metasurface pattern is written into ZEP520A-positive electron-beam resist (thickness 600 nm) using a 125 kV electron-beam lithography system. After resist development and oxygen plasma descum, the patterned holes in the electron-beam resist are backfilled with amorphous TiO2 through atomic layer deposition up to a height of 200 nm above the resist patterns. The excess TiO2 is etched back using reactive ion etching until the resist surface is exposed and the residual resist is removed by immersion in Remover PG solution for 30 h, which leaves free-standing TiO2 nanopillars or nanofins.

### Device characterization

For the 2D phase-singularity sheet, 532 nm laser light is incident normally on the non-patterned face of the metasurface and the transmitted light is imaged by a microscope with a ×100 objective. The intensity image is captured at 41 z-planes, where z = 0 mm corresponds to the patterned surface of the metasurface. At each z-plane, the intensity image is captured at six different exposure times. These multiple exposure images are weighted by their respective exposure times and stacked to remove saturated pixels and produce a composite image with a large intensity dynamic range. The field phase at each z-plane was obtained by a modified version of the ﻿single-beam multiple-intensity reconstruction technique31.

For the 2D polarization singularity sheet, 532 nm light at 45° linear polarization is incident normally on the non-patterned face of the metasurface and the transmitted light is imaged by a microscope with a ×100 objective. Between the objective and the tube lens, we place a quarterwave plate and linear polarizer to act as a polarization analyzer. At each of the 41 z-planes, we capture 36 intensity images (3 exposure times for exposure-weighted image stacking) where the quarterwave plate fast axis is rotated in steps of 5°. We measure the four unnormalized Stokes parameters for each pixel using the rotating quarterwave plate method36.

## Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## Code availability

The code that support the findings of this study are available from the corresponding author upon reasonable request.

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## Acknowledgements

We thank N. Rubin for providing the polarization-sensitive metasurface library and M. Tamagnone for assistance with developing fabrication recipes. All the aforementioned individuals are from Harvard University. S.W.D.L. is supported by A*STAR Singapore through the National Science Scholarship Scheme. J.S.P. is on leave from the Korea Institute of Science and Technology. M.L.M. is supported by NWO Rubicon Grant 019.173EN.010, by the Dutch Funding Agency NWO. This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-19-1-0135. We acknowledge financial support from the Office of Naval Research (ONR) MURI program, grant no. N00014-20-1-2450. This work was performed in part at the Harvard University Center for Nanoscale Systems (CNS), a member of the National Nanotechnology Coordinated Infrastructure Network (NNCI), which is supported by the National Science Foundation under NSF award number ECCS-2025158.

## Author information

Authors

### Contributions

S.W.D.L. conceived the algorithm, designs, and experiment. J.S.P. fabricated the samples. S.W.D.L. characterized the samples. M.L.M. and A.H.D. clarified the interpretation of the work. A.H.D. performed topological analysis of the fields. S.W.D.L., A.H.D., and F.C. wrote the manuscript. F.C. supervised the research.

### Corresponding authors

Correspondence to Soon Wei Daniel Lim or Federico Capasso.

## Ethics declarations

### Competing interests

A provisional patent application (US 63/140260) based on this work is pending. The authors declare no other competing interests.

Peer review information Nature Communications thanks Aleksandr Bekshaev, Halina Rubinsztein-Dunlop, and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Lim, S.W.D., Park, JS., Meretska, M.L. et al. Engineering phase and polarization singularity sheets. Nat Commun 12, 4190 (2021). https://doi.org/10.1038/s41467-021-24493-y

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• DOI: https://doi.org/10.1038/s41467-021-24493-y

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