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Colorful Optical Vortices with White Light Illumination

The orbital angular momentum (OAM) of light holds great promise for applications in optical communication, super-resolution imaging, and high-dimensional quantum computing. However, the spatio-temporal coherence of the light source has been essential for generating OAM beams, as incoherent ambient light would result in polychromatic and obscured OAM beams in the visible spectrum. Here, we extend the applications of OAM to ambient lighting conditions. By miniaturizing spiral phase plates and integrating them with structural color filters, we achieve spatio-temporal coherence using only an incoherent white light source. These optical elements act as building blocks that encode both color and OAM information in the form of colorful optical vortices. Thus, pairs of transparent substrates that contain matching positions of these vortices constitute a reciprocal optical lock and key system. Due to the multiple helical eigenstates of OAM, the pairwise coupling can be further extended to form a one-to-many matching and validation scheme. Generating and decoding colorful optical vortices with broadband white light could find potential applications in anti-counterfeiting, optical metrology, high-capacity optical encryption, and on-chip 3D photonic devices.

optical shot noise (35,36). Moreover, compatibility with white light illumination is important for high-volume optical authentication and optical security.
To extend the generation of optical vortices to the broader ambient lighting conditions, we increase the spatio-temporal coherence of light sources through 3D printed free-form optical components. Partly due to the limitation of dispersion engineering and time-bandwidth product in 2D photonic system (37), previous planar metasurfaces and 2D photonic crystals have only exploited OAM at single or multiple discrete wavelengths (10,23,38,39). Optical vortices that operate with incoherent and broadband ambient/white light sources are yet to be demonstrated.
On the other hand, 3D printed nanophotonics hold advantages over planar photonic systems by providing an additional degree of freedom in thickness variation (37). With 3D printed freeform optical components, we achieved the coloration of optical vortices that produce brilliant visual effects of colorful rings whose diameters depend on its OAM value or topological charges (i.e., indices for helical wavefronts).
Encoding both color and OAM information in the form of colorful optical vortices, these 3D free-form optical components act as building blocks for an optical security label on a transparent substrate. The invisibility of OAM information enhances the security towards next generation optical anticounterfeiting. The printing of these colorful rings can be achieved by spatially arranging these colorful optical vortices, and the topological charges can be examined at the designate focal plane under the illumination of broadband white light source.
Alternatively, a pair of such optical elements with complementary topological charges and matching colors will provide an avenue for encoding and decoding discretized information of topological charges and colors. Thus, color, pattern placement, and OAM information can be encoded onto a 3D printed photonic "tally" for pairwise authentication.
Instead of employing a single photonic chip with optical cryptograph, pairwise photonic devices with complementary features hold the potential for more secure optical anticounterfeiting and ease of end-user validation. However, the development of such pairwise optical anticounterfeiting labels that do not necessitate public keys or additional arithmetic is still at its infancy (32,40,41). Here, we present a new pairwise optical anticounterfeiting device composed of colorful optical vortices with information encoded in their spectrum, OAM, and spatial coordinates. We draw inspiration from the multi-dimensional complementary features first introduced in the tiger tally of ancient China, one of the earliest authentication methods in the history of security and anticounterfeiting (Supplementary Information).

Results and Discussion
The spatio-temporal coherence of light sources is represented by the complex degree of coherence ( ) 12 γ τ . It can be subdivided into the spatial 12 µ and the temporal ( ) γ τ degrees of coherence, as shown in Fig. 1(a) (see details in Supplementary Information). To date, the generation of optical vortex beam necessitates coherent laser sources, shown near the origin of the coherence diagram ( Fig. 1(a)). The resulting polychromatic and featureless focal spot prevent optical vortex beams to form from the broader region of incoherent light sources, especially for the most common and convenient ambient light source, i.e., white light illumination. To expand vortex beams to incoherent white light illumination, improvements in both spatial and temporal coherence are necessary to achieve optical vortices instead of a blurry focal spot ( Fig. 1(b)). The spiral phase plate (SPP) is miniaturized to 25 μm in diameter to achieve sufficient spatial coherence (Supplementary Information). On the other hand, the temporal coherence or the coherence length is enhanced by 6 to 13 times at by incorporating color filters in the form of nanopillars on the top surface of the SPP (Supplementary Information). As a result, colors are imparted to optical vortex beams. As shown in Fig. 1(b), our colorful vortex beam (CVB) unit operates under white light illumination and converts the blurry white focal spot into a green doughnut-shaped intensity profile, characteristic of optical vortex beams. CVBs thus convey information in the dimensions of OAM and colors. Apart from the color information and OAM topological charge, the spatial position of CVBs provide an additional degree of freedom for encoding information. Thus, a photonic tally encrypting three dimensions of information, i.e., color, OAM, and spatial position is realized.
Figure 1(c) shows the concept of the photonic tally. Photonic Tally A is the "master" or lock piece, here shown with 9 CVB units with predefined topological charges, color, and positions.
When illuminated with a broadband white light source, it generates CVBs seen as colorful doughnut shaped intensity profiles at its focal plane, where the size of the doughnut represents the magnitude of the topological charge. On the other hand, Photonic Tally B1 consists of CVBs with matching colors and positions with respect to Photonic Tally A but with conjugated topological charges. When overlaid and aligned with this photonic tally "key", the vortex beams generated by the first piece is exactly "untwisted" by the second and projects bright colorful spots (not doughnuts) at the combined focal plane of the tally pair. In addition to this one-to-one pairwise coupling between the photonic tally pair, a one-to-many matching and validation scheme of photonic tally security labels can be further exploited. For instance, when the same master tally is paired with another key, Photonic Tally B2, the image at the focal plane switches to a set of colorful rings with increasing radii, i.e., topological charge, towards the outer perimeter. Leveraging on both color and topological charge, Photonic Tally B3 is an example of a tally with some unmatched color channels in addition to topological charge, leading to colorful rings only at specific positions, while blocking the rest. The final set of patterns from B1, B2, B3 can thus be validated against a database.
To achieve a compact colorful doughnut shaped intensity at the focal plane and increase the signal to noise ratio (SNR) of the photonic tally pair (Fig. S3), we introduce a converging phase to our CVB unit. Adding a lens profile to the SPP with pillars brings the intensity profile from the farfield to the Fresnel region, allowing good alignment between the photonic tally pair and avoiding color evolution along the optical axis (39). As shown in Fig. 2(a), our high SNR CVB unit is composed of a converging phase, spiral phase, and nanopillars filter. The CVB unit thus integrates three different functions of focusing, vortex beam generating, and color filtering into a single optical element. The height profile of the CVB unit is given by , for a given focal length f and topological charge i l , designed for at the transmittance peak wavelength j λ . Here, n ∆ is the refractive index difference between photoresist IP-Dip and the surrounding air environment, r and ϕ are radial and azimuthal coordinates respectively, and 0 h is the minimum thickness of the CVB unit.
Next, nanopillars of constant height are arranged on the top surface of the CVB unit, such as to avoid placing pillars at the central singularity or lateral discontinuity to achieve a uniform arrangement of nanopillars (Fig. S4).
To speed up the two-photon polymerization (TPP) 3D printing of the photonic tally and reduce the memory consumption, we adopted a strategy of two-density three-step nanoscale 3D printing for the inner region and surface of the CVB unit (see details in Methods and Fig. S5).
Instead of directly converting 3D models into Nanoscribe's file format, we coded a script to print our CVB units in three steps. First, the inner region of the CVB unit was printed on a glass substrate at high speed and relatively rough spacing and hatching distance (Methods), providing adhesion to the substrate and shell structure of the CVB unit, which we called the low-density stage. Secondly, in the high-density stage, the top surface of the CVB unit was printed with high scanning density and slower speed, providing more accurate control of the CVB unit height profile as well as the helical wavefront (Methods). In the third step, nanopillars with constant heights were added onto the CVB unit surface in the order of height coordinates (See Supplementary Information Video 1). After the two-density three-step laser printing, the sample is developed to remove uncured resin, followed by UV-curing to strengthen the polymer nanostructures and avoid obvious shrinkage caused by incomplete curing. As shown in Fig. 2 (42) and optical security devices (43).
We designed CVB units with 25 μm diameter and 60 μm focal length to account for multiple factors, e.g., spatial coherence, numerical aperture (NA), SNR of the CVB unit, thickness, color uniformity, fabrication time, and ease of alignment of two tallies. The experimental averaged spectra ( Fig. 2(f)) and optical micrograph ( Fig. 2(g)) of the CVB unit verify these three distinct color channels, which can be recognized easily by eye or imaged with a charge-coupled device (CCD) camera followed by image postprocessing. As shown in Fig To verify the concept of pairwise authentication of the photonic tally pair, we fabricated and tested the "key" tally for the "SUTD" motif. Here, the corresponding topological charges are conjugated. These charges account for the same degree of phase wrapping but in opposite handedness. Thus, the net topological charge will be 0 after passing through a pair of conjugated CVB units. If this works, combining the tally pairs will convert the previously colorful ring array into an array of colorful dots instead. This cancellation of topological charges and conversion from topological vortex beams to simple converging beams are shown in Fig. 4(a) and (b). We show this for the two "SUTD" colored photonic tallies with OAM encrypted digital codes for "π" and spatial pattern information of "□". The corresponding "keys" with CVBs shown in the middle column will untwist the "master" optical vortices and produce To demonstrate the one-to-many matching and validation scheme, Fig. 4(a), (c) and (d) show that a single tally encrypting "π" manifests three different patterns of colorful rings at the combined focal plane when combined with different keys. With its conjugated piece "π*", the test results are purely colorful dots without topological nontriviality. However, when combined with photonic tally pieces "A" and "B", the test results show the recognizable spatial patterns of "□" and "֎" instead (Supplementary Information Video 5 and 6   (Fig. 5(a) and (b)), we can choose to completely block certain bright dots, e.g., by combining red and blue filters. Only the cascaded CVB units sharing the same color can produce discernible colorful dots when combined with its pair. In contrast, the cascaded CVB units with different subtractive colors will significantly attenuate the detected signal on the CCD camera and result in a naught in corresponding positions (Fig. 5(d)). On this principle, the "Tetris" photonic tally piece will also eliminate some Tetris pieces (not according to the actual rules of the game) when combined with its companion Tetris of different colors. As shown in Fig. 5(c) and (d), only half of the Tetris pieces remain after cascading with an altered "Tetris" photonic tally piece ( Supplementary Information Video 7). The elimination of colors and the annihilation of topological charges increase the diversity of our photonic tally pairs and improve the security level.
In conclusion, the TPP lithography enables the 3D printing of custom designed optical elements that achieve high spatio-temporal coherence to generate colorful vortex beams (CVB) under the white light illumination. When stacked and aligned in a pairwise manner, these CVB units generate and perform arithmetic addition functions of topological charges, in addition to transmission or blocking of the vortex beams with same or different colors. Encrypting spatial coordinates, color, and OAM information, our photonic tallies broaden the validation methods of optical security labels with a pairwise working principle. The resulting output patterns can be machine read and compared with a secured database for physical-digital authentication. By controlling colors and topological charge independently, we achieve 33 possible CVB states for each unit. With only a 10×10 array of such CVB units, each of our photonic tally holds up to 33 100 possible combinations, ensuring unique tallies for next-generation optical anticounterfeiting devices.

Optical simulation of the CVB unit
The simulation of nanopillars on tilted substrate is carried out using a commercial software  Fig. 3) are converted to colorful rings with information of topological charges (third column with purple edges in Fig. 3). The alignment of photonic tally pairs and corresponding optical characterization are also performed on the same microscope.

Associated Contents Online Content
Any methods, additional references, source data, extended data, supplementary information, acknowledgements; details of author contributions and competing interests; and statements of data and code availability are available online.

Data availability
The data that support the figures and other findings of this study are available from the corresponding authors upon reasonable request. Source data are provided with this paper.

Code availability
The code used for the photonic tally design and characterization is available from the corresponding author upon reasonable request.

Competing interests
The authors declare no competing financial interests.     The analogy between "tiger tally" and "photonic tally".
The tiger tally is a 2,300-year-old tiger-shaped tally from ancient China, was used to authenticate military orders. One piece was held by the military general, and the other complementary piece held by the monarch. When brought together, the two pieces would precisely match like pieces of a puzzle, as their outer shapes, inner morphology, and surface characters are complementary, as shown in Fig. S1(a). Drawing parallels to the tiger tally, we designed two 3D printed photonic tallies to generate colorful optical vortices under broadband white light illumination. This "photonic tally pair" contains conjugated topological charges at corresponding spatial positions ( Fig. S1(b)). Moreover, the colors of vortex beams also play the role of multiplexing optical information thus enhancing the security of photonic tally pair. When the photonic tally pair are brought into proximity with each other and illuminated by broadband white light, only vortex beams with specific topological charges and colors can form the pre-designed machine-recognizable patterns.
Figure S1(c) shows the design concept consisting of pairs of optical elements that form the basis of the photonic tally. The SPP is a well-known optical element to generate optical vortex carrying OAM, by introducing a spiral phase delay about its asymmetric central point.
However, if the vortex beam passes through another SPP with the opposite topological charge and same orientation of lateral discontinuity, it will effectively resemble a beam passing through the combined glass slab with negligible fluctuation in amplitude and phase of incident beam (Fig. S1(c) inset). Analogous to the complementary inner morphology of the tiger tally pair, the SPP pair has complementary morphology, thus leading to counteracting OAM in momentum space and conjugated topological charges of phase. Vortex beam with spiral phase taking the form of ( ) exp ilϕ has the twisted wavefront as shown in Fig. S1(d), where l represents topological charges of the vortex beam. The wavevector of a vortex beam is perpendicular to its equiphase surface, possessing the components along its propagating direction and azimuthal direction(1), as indicated by the colorful arrows in Fig.   S1(d) and Fig. S2. The azimuthal components of vortex wavevectors k ϕ are proportional to its OAM value and will counteract parts of each other when the optical vortices coalesce, indicated by the gradient red arrows. Thus, not only in real space does the complementary morphology of SPP pair extinguish the wavefront perturbation and abide by rectilinear propagation of light; but also in momentum space, as the angular momentum provides complementary features for the design of photonic tally pair.

The coherence analysis of CVB units.
The spatio-temporal coherence is represented by the complex degree of coherence ( ) 12 γ τ , which is defined by the mutual coherence function, ( ) ( ) ( ) In conclusion, our CVB units achieve sufficient spatio-temporal coherence and can be used directly under an incoherent white light source in a common optical microscope.

Topological charge analysis of the CVB units under broadband white light illumination.
We To analyze the colorful optical vortices under broadband white light illumination, we examined intensity and phase profiles at the neighbor wavelengths region of 457 nm, 532 nm, shown in Fig. S12, when the central wavelength of nanopillar color-filters match the design wavelengths, the intensity profiles remain doughnut shape in the whole range of high transmittance windows (Fig. S12(a)-(c)). Meanwhile, if we define the topological charges q as phase wrapping along an arbitrarily closed loop C in the doughnut shape intensity, ( )