Twisted Rainbow Light and Nature-Inspired Generation of Vector Vortex Beams

Twisted vector light beams (optical vortices) arise from a spiral modulation of the geometric (Pancharatnam–Berry) phase converting the light spin to the orbital angular momentum. The preferred geometric-phase elements using liquid crystals and plasmonic metasurfaces realize this conversion by structuring their building blocks, i.e., precisely orienting individual crystal molecules or plasmonic nanoantennas. Here, an analogous mechanism is discovered in the spiral phase modulation of light reﬂected by dielectric spheres and ﬁrst demonstrated in natural phenomena, namely in the rainbow formation. The spiral geometric phase is documented by holographic imaging of full circle primary and secondary rainbows created in the laboratory. The measurement uses a wide-angle holographic camera (ﬁeld of view ≈ 120 ° ) taking time-resolved self-correlation holograms (300 ms). The holograms allow a quantitative restoration of the spiral geometric phase of light reﬂected from thousands of randomly falling water drops. The capability of individual drops to generate vector vortex beams under circularly polarized illumination is proven theoretically and demonstrated in experiments using glass microspheres. The spherical reﬂectors are discovered as simple generators of vector vortex beams and vortex arrays, inspiring novel geometric-phase elements.


Introduction
The natural world is full of intriguing optical phenomena forming foundations of bio-inspired photonics. [1,2]This DOI: 10.1002/lpor.202200080well-established scientific field provides artificial materials springing inspiration from living organisms.Here, we first demonstrate that also optical atmospheric phenomena, namely the rainbow formation and the light reflection from water drops, show remarkable analogies with advanced geometric-phase technologies and unconventional states of light known as optical vortices.
Optical vortices are electromagnetic equivalents of twisted fields studied throughout physics.The helical wavefront and capability to carry the orbital angular momentum make the optical vortices attractive for diverse areas of optics, including optical tweezing, [3,4] optical communications, [5,6] quantum information, [7,8] phase-contrast microscopy, [9,10] optical inspection, [11,12] localization and super-resolution imaging, [13,14] or orientation imaging. [15]he optical vortices were traditionally generated by spiral phase plates, [16,17] optoelectronic devices, [18,19] and Gouy phase converters [20,21] modulating the dynamic phase of light associated with the optical path.The emerging technologies provide new possibilities for controlling light through the geometric phase discovered by Pancharatnam and Berry that is independent of the optical path. [22,23]The geometric-phase elements benefit from the broadband [24][25][26] or achromatic operation, [27,28] sensitivity to polarization states, [29][30][31] excellent integrability, [32,33] and the possibility of generating light fields with arbitrarily designed wavefront and controlled polarization. [34]Unique properties of the geometric-phase elements allow generation of vector vortex beams, [35,36] mutual conversion between spin and orbital angular momentum, [37][38][39] or preparation of structured vortex fields [40] and vortex arrays. [41,42]he generation of vector vortex beams needs a spiral modulation of the geometric phase provided by diverse but technologically demanding operations.For instance, inducing the birefringence in silica-glass by femtosecond laser writing, the radially polarized vortex converter was realized. [43,44]The noticeable progress in the spiral phase modulation and preparation of vector vortices was achieved by the fabrication of subwavelength gratings [45] and by using photoaligned liquid crystal polymers. [46,47]Recently, plasmonic metasurfaces utilizing nanofabrication and the spacevariant setting of metal or dielectric nanoantennas have brought new possibilities in creating complex vortex fields. [48,49]his paper focuses on the spiral modulation of the geometric phase of light in dielectric spheres.We show that spiral phase modulation arises from spatial variations in the reflection geometry, which result in the azimuthal rotation of the polarization ellipse.We investigate a natural geometric-phase modulation of light reflected from water drops and demonstrate the spiral phase at the points of the rainbow arc.An analogy of this effect with the mechanism of spiral modulation in polymer liquid crystals and plasmonic metasurfaces is outlined.We use the challenging selfcorrelation holographic experiments to demonstrate the twist of natural rainbow light originating from spiral variations of the geometric phase in falling water drops.The experiments build on the quantitative phase imaging and use a custom-designed wideangle holographic camera.Recorded time-resolved holograms (300 ms) allow restoring quantitative images of full circle primary and secondary rainbows in angular fields of view of ≈84°a nd ≈102°, respectively.After demonstrating the spiral geometric phase of light at the points of the rainbow arc, we focus on the light reflection in individual water drops.In experiments using glass microspheres instead of water drops, we verify that the spiral phase modulation generates white light vector vortex beams, provided the microspheres are illuminated by circularly polarized light with spatial coherence enhanced by spatial filtering.The easy and cheap fabrication of glass microspheres allows straightforward generation of microscaled vortex beams and vortex arrays.The revealed principle of geometric-phase modulation can be further elaborated to realize new types of light shaping elements.

Spiral Modulation of Geometric Phase in Spherical reflectors
The preferably used methods for controlling light through the geometric phase utilize structured optical materials, including photoaligned polymer liquid crystals and plasmonic metasurfaces.These elements modulate the geometric phase of light by structuring their basic building blocks, i.e., liquid crystal molecules and individual nanoantennas.The modulation mechanism relies on the handedness reversal of the circularly polarized light accompanied by variations of the geometric phase.The phase retardation  needed for polarization transformation stems from anisotropy caused by the elongated shape of liquid crystal molecules and nanoantennas.The change in the geometric phase is proportional to the angular orientation of photoaligned liquid crystal molecules (Figure 1A) and nanoantennas (Figure 1B); hence, their space-variant setting allows on-demand wavefront shaping independent of the optical path.When the angular position of the basic building blocks rotates and holds the angle  0 constant ( 0 = −∕4 in Figure 1), the spiral geometric phase is imposed on the incident light.Using illumination beams with the right-hand circular polarization (RHCP, the state vector |R⟩) and the left-hand circular polarization (LHCP, the state vector |L⟩), vortex beams with the helical wavefronts e ±i2 are generated, characterized by the topological charges l = ±2.The vortex l = 2 (Figure 1A) is imposed on the light transmitted through liquid crystals when anisotropy of molecules changes the |R⟩ polarization of the incident light to the |L⟩ state.A plasmonic metasurface (Figure 1B) reflects the circularly polarized light into the copolarized and cross-polarized states while imposing the geometric phase on copolarized light. [50]he amplitude of modulated light can be controlled in the specially designed metasurface, realizing a highly efficient helicitypreserving reflector. [51]The examples in Figure 1 illustrate the mechanism of spin to angular momentum conversion requiring inhomogeneous anisotropy, [39] but the implementation options are more diverse.The other approaches use cholesteric liquid crystals [52,53] and dielectric plasmonic metasurfaces [49] in reflection and transmission geometry, respectively.
This paper reveals the spiral modulation of the geometric phase of light in dielectric spherical reflectors.The dielectric sphere illuminated by the light beam with |R⟩ polarization reflects light into copolarized |R⟩ component with the geometric phase modulated azimuthally by e −i2 .The reflected light also contains |L⟩ component whose geometric phase remains unmodulated (Figure 1C).This effect is a consequence of the spacevariant polarization transformation.The parallel rays entering the dielectric sphere (Figure 1D) are reflected with Fresnel coefficients for the s and p polarization components.The ray caustic is the same in the impact plane with any azimuthal angle, meaning that the reflection keeps rotational symmetry.The Fresnel coefficients change radially on the reflecting surface.This causes that the input circular polarization is changed to elliptical (details are in Supporting Information, Part SI 1).The major axis of the polarization ellipse rotates azimuthally as shown in Fig- ure 1E.The azimuthal rotation of the polarization ellipse mimics the angular setting of the basic building blocks in structured optical materials.This analogy using Jones-matrix formalism is demonstrated.
The electric fields of the incident and reflected light are assigned to the vectors |E I ⟩ and |E R ⟩, whose components oscillate along the x-and y-axes of the Cartesian coordinate system (Figure 1C).The light reflection is described by the complex coefficients  s and  p , determining the reflection of s and p electric field components.These components oscillate along the sand p-axes of the coordinate system, which rotates with the azimuthal angle  (Figure 1C).The weight coefficients A and B depend on the magnitude and phase of the reflection coefficients where  s = | s |,  p = | p |e i for a glass-air boundary.Equations ( 1) and ( 2) describe the spiral modulation of the geometric phase of light caused by the internal reflection in dielectric spheres illuminated by the circularly polarized light.The efficiency of the spiral modulation is determined by the weight coefficient A. The modulation preserves the input circular polarization and creates the helical vortex phase e ±i2 with the topological charges l = ±2 for |L⟩ and |R⟩ input states, respectively.The light reflected with the weight B is unmodulated and the input polarization is reversed.The factors A and B depend on the incidence angle , which varies with the radial coordinate r of the reflecting surface (Supporting Information, Part SI 6).The polarization superpositions [Equations ( 1) and ( 2)] create the elliptical polarization of the reflected light with the azimuthally rotating major axis (Figure 1C).In the peripheral radial zone, Brewster´s reflection occurs ( p = 0), and the reflected light gets a linear polarization oscillating along the azimuthal direction (so-called azimuthal po-larization) given by the state vector |⟩ = (sin , − cos ) T .The vector fields obtained under the Brewster reflection for incident light with |L⟩ and |R⟩ polarizations follow from Equations ( 1) and ( 2) and can be written as The results show that the spherical reflectors can provide vector vortex beams of different polarizations and topological charges.After removing unmodulated light by polarization filtering, the spirally modulated light having the circular polarization and the pure vortex phase with l = ±2 is obtained [Equations (1) and ( 2)].The overall field created by the Brewster reflection carries the azimuthal polarization and the vortex phase with l = ±1 [Equation ( 4)].

Evidence of Spiral Geometric Phase in Natural Rainbow Light
The significance of the discovered geometric-phase spiral modulation in dielectric reflectors is increased by the frequent occurrence of spherical objects in nature.Here, the attention is focused on water drops formed by the surface tension.It is aimed to  demonstrate the spiral phase modulation in falling water drops that create the rainbow when illuminated by incoherent white light.
The water drops reflect light depending on the impact height.The maximum intensity of the emerging light is reached at the minimum deviation angle.The double light refraction in the water drops causes the minimum deviation angle Δ to be wavelength-dependent (Figure 2A), decomposing white light into the rainbow colors.Consequently, rainbow colors are perceived in minimally deviated light. [54]The outer and inner boundaries of the primary rainbow are created in red and blue colors, as shown in Figure 2A,B, along with the respective angles.Equations ( 1) and ( 2) demonstrate the occurrence of the helical vortex phase in dielectric spheres illuminated by a coherent circularly polar-ized light.When studying the rainbow in nature or mimicking these conditions in a laboratory, illumination of water drops by spectrally broadband, spatially incoherent, and unpolarized light must be considered.Hence, an overall complex field reflected from falling water drops fluctuates randomly and rapidly, and the spiral phase change in the individual drops must be restored from the self-correlation function provided by the time-resolved incoherent holograms.The proposed computational model supports the measurements performed in a phase-imaging holographic setup and explains the reconstruction of the spiral phase from the self-correlation function.
Our model describes a full circle rainbow detected by a lens with the optical axis passing through the center of the light source.In this geometry, the antisolar point [54] and rainbow center determining the origin of the polar coordinate system r,  lie at the lens optical axis.The mean rainbow radius is given by the visual angle  ( = 41, 5 • for the primary rainbow) and the distance of water drops from the lens.Since the internal reflection of light rays undergoing minimal deviation occurs close to Brewster's polarization angle, the water drops reflect only the component of the electric field.This component is perpendicular to the plane of incidence, and the light received by the lens is azimuthally polarized (Figure 2B).The degree of polarization obtained for a local linear polarization is 0.95 for primary and 0.90 for secondary rainbow. [55]The electric field of the light emanating from the rainbow points can be written as where The intensity I ′ represents a correlation record (an incoherent hologram) composed of the DC term (first two terms) and the complex conjugate cross-correlation terms The cross-correlation function Γ used for the quantitative reconstruction of the rainbow image is in experiments separated using carrier frequency provided by off-axis holography (Supporting Information, Part SI 4).The rainbow arises in the incoherent broadband illumination; hence, only light of the same frequency and coming from the same rainbow point can interfere.From a rainbow point with the azimuthal coordinate , light in |L⟩ and |R⟩ polarization is emitted, carrying the spiral phase e i and e −i , respectively.Two diffraction image spots of this point are created, overlapping on the detec-tor.As the electric fields of the images are projected to the same direction , they give rise to the interference (self-correlation) image.The cross-correlation function represents the superposition of self-correlation images of all rainbow points.The changes of the geometric phase occurring at individual rainbow points are quantitatively reconstructed from Γ.The computational model of the self-correlation holographic imaging provides results consistent with experiments.In the case of ideal imaging with an infinite system aperture, the diffraction image spots become the point images given by the Dirac delta function.In this case, the self-correlation function can be written in the form The phase image of the rainbow is defined by  = arg{Γ}, which gives  = 2 − 2.The reconstructed phase  represents the spiral geometric phase 2 whose angular orientation is determined by the polarizer setting angle .The imaging principle guarantees that the dynamic phase of the interfering waves is canceled out in the self-correlation records.Hence, the experiment is not sensitive to disturbances in the optical path, making it robust.This exceptional feature allowed measuring geometric phase variations in falling water drops (detailed discussion is in the Supporting Information, Part SI 5).
The connection between the measured complex rainbow image Γ and an optical vortex field is well documented by the numerical Fourier transform U =  {Γ}.Approximative form of the complex amplitude U is obtained considering a single wavelength  of the radiation used.In this case, |Γ| is given by the Dirac delta function used with the angular radius of the rainbow ring .The Fourier transform then results in where J 2 denotes the Bessel function of the first kind and second order.The Bessel beam has the topological charge l = 2, and the radius of the first ring of its spot is determined by the angular rainbow radius as r ′ 0 = 0.8∕ sin  (r ′ 0 = 1.2 for the primary rainbow).The vortex Bessel beam is not created when the Fourier transform is performed optically because the light is uncorrelated at different angular positions of the rainbow ring.

Experimental Strategy
In experiments, it was focused on measuring the spiral phase of the rainbow light examined in the phase imaging model.A custom-built wide-angle holographic camera was used whose optical setup is shown in Figure 2D.The measurements required a rainbow formed in the laboratory conditions and experimental geometry suitable for the full circle rainbow imaging.The collimated light of the discharge lamp replacing sunlight (not shown in the scheme) illuminated a cloud of water drops created by a water spray.At the same time, the holographic imaging system was placed between the light source and the falling water drops.The antisolar point and the rainbow center lay at the system optical axis passing through the center of the light source, providing symmetry for full circle rainbow measurement.Detection of the primary and secondary full circle rainbows needs light collection in a cone with an angle greater than 105°.To fulfill this requirement, an imaging system comprising of the microscope objective MO (40 ×, NA = 0.95), the lens L 1 (f = 200 mm), and the camera lens CL 1 (f = 50 mm) was designed.The MO with a short focal length acted as a fisheye lens, collected light and created an intermediate rainbow image inside the MO.The lens L 1 collimated light to prevent angular spreading, while the lens CL 1 created a circle rainbow image (Supporting Information, Part SI 3).The correlation imaging of the rainbow spiral phase, using a holographic module attached at the output of the wide-angle imaging system (Figure 2D) was performed.This module was designed as a common-path interferometric system, which uses polarization control of light to record off-axis incoherent holograms. [56,57]he module combines Leith's and Upatnieks' implementation of an achromatic off-axis interferometer [58] with operation of a geometric phase grating (GPG) (Supporting Information, Part SI 4).The GPG is placed at the image plane of the lens CL 1 and acts as a polarization-sensitive diffractive beamsplitter directing |R⟩ and |L⟩ components of the rainbow light into −1st and +1st diffraction orders.Camera lens CL 2 (f = 50 mm) captures the diffracted light and together with lens L 2 (f = 200 mm) projects the GPG plane at the sCMOS camera (Andor Zyla 4.2).The linear polarizer P placed between CL 2 and L 2 allows interference of the waves at the sCMOS camera.The diffractive dispersion of the GPG causes the light of individual wavelengths reaches the detector with different inclination angles and creates an off-axis hologram with achromatic fringes. [58,59]The cross-correlation function [Equation ( 7)] restoring the spiral phase change at the points of the rainbow arc is available from the off-axis hologram adopting Fourier techniques. [59]he following experiments demonstrated the spiral modulation of the geometric phase in a single glass microsphere.The measurement used again the holographic module shown in Fig- ure 2D but attached at the output port of a commercial microscope Nikon Eclipse L150 (GPG was placed at the microscope image plane).The same optical microscope without holographic module was used to measure vector vortex beams generated by glass microspheres.The pinhole filtered the light from a spectrally broadband source in these experiments.Thanks to the enhanced spatial coherence, the spiral modulation in the glass microsphere formed a helical phase of an overall reflected light, giving rise to a vortex beam.

Experimental Demonstration of Twisted Natural Rainbow Light
We performed experiments demonstrating the spiral geometric phase of the rainbow light using the holographic setup described above (Figure 2D).The initial measurement verified the rainbow formation without using a holographic module.The sCMOS camera placed directly in the image plane of the lens CL 1 captured the intensity image (Figure 3A).The arrows in Figure 3A indicate primary and secondary rainbows with Alexander's dark band in-between and supernumerary bows, respectively. [54]The presence of ancillary phenomena well known from natural rainbow observations confirmed the correctness of our experimental procedure.Subsequently, the holographic module was connected to the wide-angle imaging system.The light of the discharge lamp created the primary and secondary full circle rainbows, and the holographic module recorded incoherent holograms on the sCMOS camera.Thanks to an off-axis geometry provided by GPG, the cross-correlation function Γ became separable by Fourier filtering and allowed amplitude and phase reconstruction (Supporting Information, Part SI 4).The amplitude and phase images of the rainbow corresponding to |Γ| and  = arg(Γ) are presented in Figure 3B,C.The amplitude and phase images in Figure 3B,C are quantitative reconstructions of a long exposure hologram (acquisition time 8 s), integrating light from water drops steadily falling through the field of view.We used a long exposure time to measure also the secondary rainbow, which is much fainter than primary.This was possible because the measured geometric phase remains unchanged during the long exposure.The movement of the water drops did not affect the recording because our method is inherently insensitive to dynamic phase disturbances related to the optical path [Equation (7), Supporting Information, Part SI 5].The arrows in Figure 3B,C indicate the secondary rain-bow in both amplitude and phase holographic images.The phase image in Figure 3C demonstrates the spiral change of the geometric phase measured at the primary and secondary rainbows.The geometric phase changes linearly with the azimuthal angle according to e i2 .Two phase jumps from − to + correspond to the difference of the topological charges Δl = 2. Their angular position is determined by the orientation angle  of the linear polarizer.The measured phase distribution is in accordance with Equation (7) following from the calculation model (Supporting Information, Part SI 2).To estimate the accuracy of the rainbow phase measurement, we evaluated the phase levels of the reconstructed and theoretical spiral phase map pixel by pixel.In this way, the standard deviation  = 0.5 rad was obtained.The image spot | {Γ}| in Figure 3D was obtained by the numerical Fourier transform of the complex rainbow image Γ shown in Figure 3B,C.This spot corresponds well to the vortex Bessel beam J 2 given by Equation (8). Figure 3E,F shows time-resolved amplitude and phase images of a rainbow originating from the spatially bounded cloud of water drops made by single spraying.The images reconstructed from holograms taken with a time step of 0.3 s demonstrate the gradual rainbow formation as water drops pass through the field of view.When the cloud of water drops enters or leaves the field of view, only part of the rainbow arc is created (arrows in images taken at 0 and 1.5 s).A full circle rainbow arises at 0.6 and 0.9 s when water drops fill the entire field of view.The spiral geometric phase remains apparent in all-time instances.The time-resolved images exhibit reduced contrast due to low signal-to-noise ratio but well document the formation of incoherent holograms by point self-correlation of light.

Measurement of Spiral Geometric Phase in Glass Microspheres
The spiral phase modulation demonstrated in the rainbow stems from the effects occurring in individual water drops acting as dielectric spheres.Here, our interest focuses on changes in the geometric phase of light in the entire reflective area of the glass microspheres.We aimed to verify the spiral modulation beyond the azimuthal polarization approach justified in the rainbow study.According to theory [Equations ( 1) and ( 2)], the spiral phase modulation occurs in radial zones of the microsphere, where the polarization of the reflected light is determined by an ellipse with the major axis rotating azimuthally (Figure 1E).We tested this ability experimentally using glass microspheres with the diameter varying in the range of 5-20 μm.A mixture of microspheres with various sizes was used as a sample in the microscope Nikon Eclipse L150 with the attached holographic module.4C demonstrates the pure spiral modulation creating the geometric phase e i2 in the whole reflection area.This result proves that spiral modulation is not conditioned by the azimuthal polarization and occurs even if the reflected light has a spacevariant elliptical polarization (Figure 1E).In this case, the spiral geometric-phase modulation is caused by the rotation of the polarization ellipse, whose major axis maintains the azimuthal direction.Hence, the spiral phase patterns are identical for all microspheres and have the same angular orientation given by the setting angle  of the output polarizer.At the microsphere center, the orientation of the major axis is undefined, causing the geometric phase is singular and |Γ| = 0 at that position.The bright rings reconstructed in the amplitude image represent the full circle rainbows formed by the individual microspheres (Figure 4D). Figure 4E,F demonstrates the phase map and the amplitude spot of a Bessel beam [Equation (8)] obtained by the Fourier transform applied to the complex image Γ of the microsphere indicated by a square in Figure 4D.

Demonstration of Vector Vortex Beams from Glass Microspheres
The holographic measurement of glass microspheres performed with the incoherent and unpolarized light demonstrated the spi- ral change of the geometric phase at individual points of the reflection area.In further experiments, we focused on the light propagation after reflection from the microsphere.We used a collimated light beam approximating a plane wave to prove the generation of vector vortex beams from glass microspheres.In spatially coherent light, the spiral geometric-phase modulation was combined with the dynamic phase shaping, which created a focused vortex beam.Equations ( 1) and (2) describe the light field emerging from the spherical reflector illuminated by a coherent and circularly polarized light beam.Experiments were performed using spectrally broadband light from a discharge lamp that was spatially filtered by a pinhole (Thorlabs P10K, diame-  Experimental demonstrations of the vortex beam and its interference with the unmodulated wave match well the results of our computational model.The interfering light fields were calculated in the Fresnel diffraction approach using the pupil function of the microsphere comprising spatially varying reflection conditions, spiral modulation of the geometric phase and shaping of the dynamic phase under influence of the spherical aberration (calculation model is available in the Supporting Information, Part SI 6).In Figure 5B-D

Conclusion
The rainbow formation is a fascinating optical phenomenon investigated in various optics approaches.Our study focuses on the spiral variations of the geometric phase resulting in the twisting of the natural rainbow light, a phenomenon that has not yet been revealed.We elucidate the mechanism of the spiral phase modulation of the rainbow light and outline its analogy with principles used in polymer liquid crystals and plasmonic metasurfaces for controlling and shaping light.The examined effects were demonstrated on full circle primary and secondary rainbows using a wide-angle holographic camera.The changes of the light phase in the falling water drops forming the rainbow were quantitatively reconstructed from incoherent holograms.Inspired by polarization effects in water drops, we developed the concept of vortex beam generation by dielectric reflectors.The experiments demonstrated the generation of vector vortex beams in glass microspheres and the formation of double-helix interference patterns known from localization microscopy and orientation imaging.The presented findings might inspire the design of new optical elements that simultaneously shape the geometric and dynamic phase of light.

2 )
The polarization states of the reflected and incident light are related as |E R ⟩ = T|E I ⟩, where T = R (∕2−) WR (−∕2) V. W and V are 2 × 2 diagonal matrices (W 11 =  s , W 22 =  p , | s | > | p | and V 11 = −1, V 22 = 1) describing the Fresnel reflection in the local coordinate system with the s-and p-axes and the back propagation of the reflected light.The matrix R rotates the coordinate system by angles ±( − ∕2).The polarization of the reflected light can be decomposed into the base of RHCP and LHCP waves with the state vectors |R⟩ and |L⟩.When dielectric spheres are illuminated by LHCP wave, |E I ⟩ = |L⟩, the polarization state of the reflected light is given by (Supporting Information, Part SI 1) |E R ⟩ = Ae i2 |L⟩ + B|R⟩ (1) Using RHCP illumination, |E I ⟩ = |R⟩, the polarization of the reflected light is changed to |E R ⟩ = Ae −i2 |R⟩ + B|L⟩ (Laser Photonics Rev. 2022, 16, 2200080

Figure 1 .
Figure 1.Spiral geometric-phase modulation by structured optical materials and spherical reflectors.A) Polymer liquid crystals: Polarization state |R⟩ of incident light changes to |L⟩ state, and the geometric phase of transmitted light varies in dependence on the angular orientation of molecules (spiral phase e i2 is obtained for constant angle  0 ).B) Plasmonic metasurface: Incident light with |R⟩ polarization (green color) is reflected as phase modulated light preserving input polarization (spiral phase e −i2 , red color) and unmodulated light with reversed polarization (blue color).C) Spherical reflector: Geometric phase is modulated through azimuthal changes in reflection geometry, resulting in the space-variant polarization of the reflected light.Incident light with |R⟩ polarization (green color) is reflected in phase modulated copolarized component (spiral phase e −i2 , red color) and unmodulated component with reversed polarization (blue color).The weight coefficients of the polarization components A and B vary radially across the reflection area.D) Nonparaxial ray tracing of parallel rays reflected from the dielectric sphere.E) Azimuthal rotation of the polarization ellipse of the reflected light.

Figure 2 .
Figure 2. Formation of a full circle rainbow and measurement of the spiral phase of rainbow light.A) Reflection of red and blue rays in water drops with the illustration of the minimum angular deviation Δ, the visual angle , and the Brewster polarization angle .B) Azimuthal polarization of light at the points of the rainbow ring.C) Decomposition of the azimuthally polarized rainbow light to |R⟩ and |L⟩ states with helical phases given by the topological charges l = −1 and l = +1.D) Experimental scheme of a wide-angle holographic camera: MO -microscope objective, L 1 -first collimating lens, CL 1 -first camera lens, GPG -geometric-phase grating, CL 2 -second camera lens, P -linear polarizer, L 2 -second lens, and sCMOS -Complementary Metal-Oxide-Semiconductor camera.Top cutout: Deflection of |R⟩ and |L⟩ polarizations to +1st and −1st diffraction orders by GPG.
S denotes the reflection coefficient for the s component, |⟩ represents the state vector of the azimuthal polarization, and |E I ⟩ is the spectrally broadband incident light given as a superposition of mutually uncorrelated monochromatic components (detailed calculation model is in Supporting Information, Part SI 2).The azimuthal polarization can be decomposed as |⟩ = (i∕ √ 2)(e −i |R⟩ − e i |L⟩), showing that the geometric spiral phase with the topological charges l = ±1 is imposed on |L⟩ and |R⟩ states in each angular position  of the rainbow.The rainbow images created in |L⟩ and |R⟩ polarization components are transformed by a linear polarizer with the orientation angle , providing an overall complex field denoted as |E ′ ⟩ = |E ′ 1 ⟩ + |E ′ −1 ⟩.The detected intensity I ′ = ⟨E ′ |E ′ ⟩ is written as

Figure 3 .
Figure 3. Quantitative amplitude and phase images of primary and secondary full circle rainbows reconstructed from incoherent holograms acquired by the wide-angle holographic camera in Figure 2D.A) Intensity snapshot verifying phenomena observed in the rainbow created in nature (Alexander´s dark band, supernumerary bows).B) Amplitude holographic image of primary and secondary (indicated by the arrow) rainbows.C) Quantitative phase image demonstrating the spiral geometric phase e i2 in the ring of primary and secondary (indicated by the arrow) rainbows.D) Amplitude beam spot | {Γ}| (approximation to Bessel beam J 2 ) obtained by the Fourier transform of the complex rainbow image in (B) and (C).Time-resolved quantitative amplitude E) and phase F) images (time step 0.3 s) showing the individual stages of rainbow formation when the spatially bounded cloud of water drops passes through the camera field of view.Only portions of the rainbow arc are created at times 0 and 1.5 s (indicated by arrows) when water drops enter and leave the field of view.A full circle rainbow arises at 0.6 and 0.9 s when water drops fill the entire field of view.

Figure
4A shows an off-axis hologram recorded under Köhler illumination with a discharge lamp.Interference fringes creating a spatial carrier frequency are apparent from Figure 4B representing enlarged detail of the square area marked in Figure 4A. Figure 4C,D shows quantitative phase and amplitude images reconstructed from the cross-correlation function Γ.The phase image in Figure

Figure 4 .
Figure 4. Experimental demonstration of the spiral geometric phase of light reflected from glass microspheres.A) Off-axis hologram of glass microspheres acquired under Köhler illumination using a discharge lamp.B) Interference fringes creating the spatial carrier frequency [enlarged square area in (A)].Quantitative phase C) and amplitude D) images reconstructed from hologram recorded under incoherent Köhler illumination.The phase map E) and the amplitude spot F) of a Bessel-like beam obtained applying the Fourier transform to the complex image of the microsphere shown in (C) and (D) (marked by the square).

Figure 5 .
Figure 5. Experimental demonstration of the vector vortex microbeam and the double-helix interference pattern generated using a glass microsphere illuminated by spatially coherent light with |L⟩ polarization.A) Intensity images of microspheres illuminated by broadband light with spatial coherence enhanced by pinhole filtering.B) Interference patterns created by the|L⟩ component (helical phase with the topological charge l = 2) and the unmodulated |R⟩ component of the light reflected from the microsphere marked in (A) (interference obtained using a linear polarizer).C) Intensity spots of the vector vortex microbeam (|L⟩ polarization, l = 2) separated from the reflected light by the polarization filtering.D) Interference pattern from A) obtained with the defocusing Δz = 6.5 μm creating a double-helix point spread function rotating with the orientation angle  of the polarizer.The insets at the top right in (B)-(D) represent theoretical results.
ter 10 μm) to enhance spatial coherence.The measurement is also possible with laser light, but the low temporal coherence of broadband light provides cleaner images free of coherence artifacts.The beam illuminating the glass microsphere was collimated and converted to |L⟩ polarization state using a circular polarizer.According to Equation (1), the reflected light is composed of a spirally modulated beam maintaining |L⟩ polarization and unmodulated light with |R⟩ polarization.The amplitude weights of the polarization components A and B vary radially across the reflecting area and show that the |L⟩ and |R⟩ components are dominantly reflected in different regions (FigureS3, Supporting Information).In addition to the spiral geometric phase imposed on |L⟩ component, all the reflected light is altered in the dynamic phase related to the optical path in the dielectric sphere.The dynamic phase alteration creates a spherical wavefront resulting in the focusing of light but also introduces a strong spherical aberration.The spherical aberration is significantly different in the reflective regions of the |L⟩ and |R⟩ beams, causing them to be focused at different distances from the microsphere.Hence, the intensity pattern created by the beam interference changes with the defocusing Δz (detailed discussion is in the Supporting Information, Part SI 6).Light fields from the glass microspheres recorded under various conditions are shown in Figure5. Figure 5A presents intensity images of microspheres captured in broadband light filtered by the pinhole.In the following experiment, the light reflected by one of the microspheres [marked by a square in (A)] was transformed by a polarizer, which allowed the |L⟩ and |R⟩ light components to interfere.Figure 5B shows the recorded interference patterns.The most pronounced interference pattern corresponds to the position Δz = 5 μm, where both interfering beams are acceptably focused.The spiral interference pattern well doc-uments the interference of the vortex beam (related to |L⟩ polarization component) with the unmodulated spherical wave (related to |R⟩ polarization component).The two arms of the spiral pattern show that the helical wavefront of the vortex beam carries the topological charge l = 2, in accordance with Equation (1).For Δz = 0, the unmodulated spherical wave is sharply focused and dominates in the interference pattern.Increasing the defocusing to Δz = 7.5 μm, the diffraction spot of the unmodulated spherical wave spreads from the Airy disk to a ring and the interference pattern is dominantly affected by the vortex beam (Figure S4, Supporting Information).When demonstrating the vortex beams, the image spots were recorded using a circular polarizer.In this case, the modulated wave with |L⟩ polarization was detected, while the unmodulated light possessing |R⟩ polarization was removed.The intensity spots in Figure 5C represent the obtained circularly polarized vortex beam with l = 2 [Equation (1)].The interference patterns in Figure 5D were obtained under the same conditions as the images in Figure 5B and differ only by the defocusing.The value of Δz = 6.5 μm was set to create a two-lobe pattern (DH PSF -Double-Helix Point Spread Function).The DH PSF typically comes from the interference of two beams with the difference in the topological charges |Δl| = 2.As the orientation of the polarizer  changes, the DH PSF rotates, as shown in Figure 5D.The symmetry of the DH PSF maintained during rotation verifies the high quality of the vortex beam generated by the glass microsphere.
, the simulation results are shown as the insets at the top right.A comparison of our results with previously published data obtained with artificial materials confirms the high quality of the spiral modulation in spherical reflectors and its analogy with Pancharatnam-Berry metasurfaces and liquid crystal components (cf.Figures5B and 3in ref.[35] or Figures 5D and 4 in ref.[60]).