Geometric Pancharatnam-Berry Phase Research Articles

Flexibly control of vortex beams based on all-dielectric Pancharatnam-Berry phase transmission encoding metamaterials

Because the ohmic loss of metal materials affects the transmission efficiency of metasurface, we design a transmission-type all-dielectric coded metasurface element structure based on the principle of Pancharatnam-Berry geometric phase. We digitally encode these cell structures to construct a coded metasurface. In order to flexibly regulate the transmission angle of vortex beam encoded metassurface, we introduce the Fourier convolution principle in digital signal processing to perform the Fourier convolution addition operation for two different coding sequences. A new coding sequence is constructed by Fourier convolution operation of digital coding unit. Through this operation, the number of elements on the coded metasurface is constant, and the scattering angle of the vortex beam is flexibly controlled. In order to realize the multifunctional integration of phase vortex beams, we introduce the principle of digital coding addition in the complex domain to add two digitally coded metasurface sequences with different functions to realize the multifunctional superposition.

Broadband convolutional scattering characteristics of all dielectric transmission Pancharatnam–Berrygeometric phase metasurfaces

In order to obtain the broadband scattering characteristics, we propose a superperiodic cell structure with all-dielectric material to construct Pancharatnam–Berry geometric phase encoding metasurfaces. Because we cannot design or prepare infinitesimal coding unit particles, according to the generalized Snell’s law, we can only obtain discrete scattering angle regulation for the basic coding metasurface sequence. In order to obtain multi-angle scattering characteristics, we introduce the Fourier convolution principle in digital signal processing on the Pancharatnam–Berry geometric phase encoding metasurfaces. By using the addition and subtraction operations on two encoding metasurface sequences, a new encoding metasurface sequence can be obtained with different deflection angle. Fourier convolution operations on the encoding metasurfaces can provide an efficient method in optimizing encoding patterns to achieve continuous scattering beams. The addition and subtraction methods are also applicable to the checkerboard coding mode. The combination of Fourier convolution principle and Pancharatnam–Berry phase coded metasurface in digital signal processing can realize more powerful electromagnetic wave manipulation capability.

The geometric phase made simple

We give a simple description of the Pancharatnam-Berry geometric phase and some of its applications in optics. Geometric phases are a universal phenomenon, but we focus here on the case of geometric phases caused by changes in polarization. The geometric origin of this phase is explained by analogy with the motion of an imaginary creature living on a small planet, inspired by Saint-Exupéry’s “The Little Prince”.

Control of phase, polarization, and amplitude based on geometric phase in a racemic helix array

The Pancharatnam–Berry geometric phase has attracted great interest due to the elegant phase control strategy via geometric transformation of optical elements. The commonly used geometric phase is associated with circular polarization states. Here, we show that by exploiting the geometric phase associated with the two elliptical eigen-polarization states in a racemic metallic helix array, exotic features including full range phase modulation for linear polarization states, diverse polarization conversion, and full complex amplitude modulation can be obtained with rotation of the helices. As a proof of concept, several devices for implementing polarization conversion, vortex beam generating, and lateral dual focusing are built with a racemic helix array in the microwave regime. The calculated and experimental results validate our proposals, which can stimulate various advanced metadevices.

Reconfigurable plasmonic nanoslits and tuneable Pancharatnam-Berry geometric phase based on electromechanical nano-kirigami [Invited

Reconfigurable three-dimensional (3D) nanostructures possess additional spatial freedom and rich physical characteristics compared with their two-dimensional (2D) counterparts. Here we demonstrate a facile and automated nano-kirigami method to create the 3D deformed spiral metasurfaces, which can flexibly manipulate optical waves by simply applying external voltages. Through etching Archimedean spirals into Au/SiO2/Si layer, multi-order localized surface plasmon resonances are excited within the spiral nanoslits and are dynamically tuned by floating the nanopatterns and employing the electrostatic forces, which induces a reflection modulation contrast as high as 189%. The 2D-to-3D transformation induces a strong modification of the optical chirality in both aspects of spatial distributions and strength. Importantly, the Pancharatnam-Berry (PB) geometric phase can be carefully designed and dynamically broken in the 2π range by rotating the direction of the spiral units, which results in the dynamic construction and erasing of metasurface holograms. The proposed scheme of tuning plasmonic nanoslits and breaking PB geometric phase by employing the electromechanical deformation provides a new degree of freedom for reconfigurable photonic systems, as well as dynamic applications in quasi-flat optical platforms.

Enhancement of efficiency on the Pancharatnam-Berry geometric phase metalens in the terahertz region.

Traditional terahertz lenses face high thickness, low transmittance, difficult processing, and other problems that are not conducive to mass production and integration. Here, we propose a wideband all-dielectric Pancharatnam-Berry geometric phase cell structure to construct a metasurface flat lens. However, when the geometrical phase element structure rotates, the transmission efficiency of the periodic element structure obviously decreases, which will lead to the decrease of the efficiency of the designed flat lens. In order to improve the efficiency, we propose to add a layer of tapered microstructure on the flat substrate to greatly improve the transmission efficiency of the element structure, thus leading to the improvement of the efficiency of the metasurface lens. By comparing the metasurface lens with conical and planar substrates, the metasurfaces with conical structure can greatly improve the transmission efficiency at broadband and wide angle ranges.

Generalized Pancharatnam-Berry Phase in Rotationally Symmetric Meta-Atoms.

Pancharatnam-Berry geometric phase has attracted enormous interest in subwavelength optics and electromagnetics during the past several decades. Traditional theory predicts that the geometric phase is equal to twice the rotation angle of anisotropic elements. Here, we show that high-order geometric phases equal to multiple times the rotation angle could be achieved by meta-atoms with highfold rotational symmetries. As a proof of concept, the broadband angular spin Hall effect of light and optical vortices is experimentally demonstrated by using plasmonic metasurfaces consisting of space-variant nanoapertures with C2, C3, and C5 rotational symmetries. The results provide a fundamentally new understanding of the geometric phase as well as light-matter interaction in nanophotonics.

Photonic Rashba effect.

The intercalation of an antennae array with a geometric Pancharatnam–Berry phase into a defective two-dimensional photonic crystal slab enables a spin-dependent splitting of directional emission in momentum space, that is, a Rashba effect for photons.

Measurement of the Pancharatnam–Berry phase in two-beam interference

Young’s dual-pinhole interference experiment with arbitrary fully correlated and polarized vector light fields leads to a Pancharatnam–Berry geometric phase that is related to the associated dynamical phase. We demonstrate theoretically and experimentally how the dynamical phase across the interference pattern can be deciphered from the total phase, thereby leaving only the geometric phase. Our results constitute the first genuine interferometric phase measurements that yield the Pancharatnam–Berry geometric phase in Young’s two-beam interference setup.

Hybrid order Poincaré spheres for Stokes singularities.

Hybrid order Poincaré spheres to represent more general Stokes singularities are presented. Polarization singularities form a subset of Stokes singularities, and therefore induction of these spheres brings completeness. The conventional understanding of Poincaré beams as hybrid order Poincaré sphere beams is also expanded to include more beams. Construction and salient properties of these spheres are explained with illustrations to show their ability to represent more exotic Poincaré beams that have zero total helicity irrespective of their size. Pancharatnam-Berry geometric phase formulation using these new spheres is also possible.

Full‐Color Complex‐Amplitude Vectorial Holograms Based on Multi‐Freedom Metasurfaces

AbstractPhase, polarization, amplitude, and frequency represent the basic dimensions of light, playing crucial roles for both fundamental light–material interactions and all major optical applications. Metasurfaces have emerged as a compact platform to manipulate these knobs, but previous metasurfaces have limited flexibility to simultaneous control them. A multi‐freedom metasurface that can simultaneously and independently modulate phase, polarization, and amplitude in an analytical form is introduced, and frequency multiplexing is further realized by a k‐space engineering technique. The multi‐freedom metasurface seamlessly combines geometric Pancharatnam–Berry phase and detour phase, both of which are frequency independent. As a result, it allows complex‐amplitude vectorial hologram at various frequencies based on the same design strategy, without sophisticated nanostructure searching of massive geometric parameters. Based on this principle, full‐color complex‐amplitude vectorial meta‐holograms in the visible are experimentally demonstrated with a metal–insulator–metal architecture, unlocking the long‐sought full potential of advanced light field manipulation through ultrathin metasurfaces.

Single-shot measurement of the space-varying polarization state of light through interferometric quantification of the geometric phase

Light beam carrying spatially varying state of polarization generates space varying Pancharatnam-Berry geometric phase while propagating through homogeneous anisotropic medium. We show that determination of such space varying geometric phase provides a unique way to quantify the space varying polarization state of light using a single-shot interferometric measurement. We demonstrate this concept in a Mach-Zehnder interferometric arrangement using a linearly polarized reference light beam, where full information on the spatially varying polarization state is successfully recovered by quantifying the space varying geometric phase and the contrast of interference. The proposed method enables single-shot measurement of any space varying polarization state of light from the measured interference pattern with a polarized reference beam. This approach shows considerable potential for instantaneous mapping of complex space varying polarization of light in diverse applications, such as astronomy, biomedical imaging, nanophotonics, etc., where high precision and near real-time measurement of spatial polarization patterns are desirable.

Pancharatnam-Berry geometric phase memory based on spontaneous parametric down-conversion.

Phase memory is an effect in which the interaction between a coherent pump beam and a nonlinear crystal generates photon pairs via the spontaneous parametric down-conversion process, then the down-converted photons (signal and idler) can carry the phase information of the pump beam. There has been much research on the memory of the dynamic phase so far; however, there is no report on the memory of non-dynamic phase, to the best of our knowledge. Here we acquire a Pancharatnam-Berry (PB) geometric phase in a physical system when light travels along a trajectory in polarization-state space. Induced coherence occurs in a cascaded scheme composed of two nonlinear crystals, when the idler photons in both crystals are aligned to be indistinguishable. A NOON ($N\; = \;{2}$N=2) state is established when blocking the two idler photons. We explore the PB geometric phase memory of the NOON state and induced coherence. We find that the first-order interference of the two-photon state or signal photons can be controlled by introducing the PB geometric phase to the pump light. This may facilitate precise control of the phase of the down-converted photons.

Localized geometric phase analogue associated with Poincaré beams due to unfolding of fractional optical vortex beams through a birefringent crystal

Fractional optical vortex beam is generated by the diffraction of a Gaussian beam using computer generated hologram embedded with mixed screw-edge dislocation. Unfolding of the generated fractional vortex beam into eigen-polarization components with orthogonal polarization results in the conversion of scalar phase singularity to vector polarization singularities in the beam cross-section. The evolution of the singularities of the ellipse field namely C-points (points of undefined major axis) and L-lines (lines of undefined handedness) in the state of polarization distribution on a transverse plane quantifies the transformation. The effect of the phase morphology dictated by the fractional order of the dislocation, transverse spatial separation and longitudinal relative phase of the two eigen-polarization components on determining the complex transverse polarization structure is investigated. The nature of the generated Poincaré beam is also indicated by projecting the states of polarization on to the Poincaré sphere. With increasing order of dislocation from 0.0 to 1.0 in fractional steps and with increasing relative phase, the partial Poincaré beam is transformed to a full Poincaré beam. The transformation of the local structure around the C-points is measured through a localized phase parameter defined similar to Pancharatnam–Berry geometric phase by the projected Poincaré sphere contour for different dynamic phase difference of the unfolded beams generated from different fractional order dislocation.

Geometric phase in beating of light waves

Beating is a simple physical phenomenon known for long in the context of sound waves but remained surprisingly unexplored for light waves. When two monochromatic optical beams of different frequencies and states of polarization interfere, the polarization state of the superposition field exhibits temporal periodic variation—polarization beating. In this work, we reveal a foundational and elegant phase structure underlying such polarization beating. We show that the phase difference over a single beating period decomposes into the Pancharatnam–Berry geometric phase and a dynamical phase of which the former depends exclusively on the intensities and polarization states of the interfering beams whereas the sum of the phases is determined solely by the beam frequencies. Varying the intensity and polarization characteristics of the beams, the relative contributions of the geometric and dynamical phases can be adjusted. The geometric phase inherent in polarization beating is governed by a compact expression containing only the Stokes parameters of the interfering waves and can alternatively be obtained from the individual beam intensities and the amplitude of the intensity beats. We demonstrate both approaches experimentally by using an interferometer with a fast detector and a specific polarimetric arrangement. Polarization beating has a unique character that the geometric and dynamical phases are entangled, i.e. variation in one unavoidably leads to a change in the other. Our work expands geometric phases into a new domain and offers important novel insight into the role of polarization in interference of electromagnetic waves.

Pancharatnam-Berry phase in electromagnetic double-pinhole interference

Young's two-beam interference and the Pancharatnam-Berry geometric phase constitute two landmark results in optical physics. Employing the spectral interference law of electromagnetic waves, we establish a general expression for the Pancharatnam-Berry phase in Young's two-pinhole setup. We show that the phase is elegantly specified by the electric field Stokes parameters at the openings. The geometric phase can also be deduced from the interference pattern using Pancharatnam's definition of the generalized phase, as we demonstrate both numerically and experimentally. Our results imply a way to control the phase differences in two-beam electromagnetic interference by adjusting the input-field intensities and polarization states.

Visible-wavelength metalenses for diffraction-limited focusing of double polarization and vortex beams

In recent years, two-dimensional functional optical devices utilizing metasurfaces to manipulate phases of light develop rapidly. In this paper, we demonstrated metalenses according to the geometric Pancharatnam–Berry phase concept based on zinc sulfide (ZnS) material. Metalenses at the visible wavelengths λ=405, 532 and 633 nm have the capacity to focus incident light down to diffraction-limited spots with corresponding transfer efficiencies of 72%, 65%, 67%, respectively. A full-hybrid metalens was developed to adapt for double-polarized light focusing, which altered the condition that metalenses designed by geometric phase concepts could not focus different polarized light into one spot. In addition, we realized the conversion of the circularly polarized plane light into a vortex beam and focusing the vortex beam simultaneously by one device.

Rotational Doppler shift induced by spin-orbit coupling of light at spinning metasurfaces

The interaction of light with spinning objects can lead to a frequency shift as a result of the transferred angular momentum. Subwavelength plasmonic nanostructures on an optical surface provide an efficient platform for local light manipulation and thus can be utilized for such momentum exchanges. Here we demonstrate reflective-type plasmonic metasurface q-plates based on the geometric Pancharatnam-Berry phase that are capable of changing the spin and orbital angular momentum of light. When these metasurfaces rotate at a constant angular speed, we observe a rotational Doppler frequency shift of the reflected light, which depends on the total angular momentum transfer. The flexibility in the design of the metasurfaces even enables complex reflective phase masks that can transfer different orbital angular momenta, at the same time, to the beam. Our experiments show that such complex metasurface phase masks can be used to obtain spatially variant Doppler shift in the reflected beam profile.

Flat polarization-controlled cylindrical lens based on the Pancharatnam–Berry geometric phase

The working principle of ordinary refractive lenses can be explained in terms of the space-variant optical phase retardations they introduce, which reshape the optical wavefront curvature and hence affect the subsequent light propagation. These phases, in turn, are due to the varying optical path length followed by light at different transverse positions relative to the lens center. A similar lensing behavior can, however, be obtained when the optical phases are introduced by an entirely different mechanism. Here, we consider the ‘geometric phases’ that arise from the polarization transformations occurring in anisotropic optical media, named after Pancharatnam and Berry. The medium anisotropy axis is taken to be space-variant in the transverse plane and the resulting varying geometric phases give rise to the wavefront reshaping and lensing effect, which however also depends on the input polarization. We describe the realization and characterization of a cylindrical geometric-phase lens that is converging for a given input circular-polarization state and diverging for the orthogonal one, which provides one of the simplest possible examples of optical elements based on geometric phases. The demonstrated lens is flat and only a few microns thick (not including the supporting substrates); moreover, its working wavelength can be tuned and the lensing can be switched on and off by the action of an external control electric field. Other kinds of lenses or more general phase elements inducing different wavefront distortions can be obtained by a similar approach. Besides their potential for optoelectronic technology, these devices offer good opportunities for introducing college-level students to an advanced topic of modern physics, such as the Berry phase, with the help of interesting optical demonstrations.

Geometric spin Hall effect of light with inhomogeneous polarization

The spin Hall effect of light originates from spin-orbit interaction of light, which manifests two types of geometric phases. In this paper, we report the observation of a geometric spin Hall effect by generating a light beam with inhomogeneous polarization distribution. Unlike the previously reported geometric spin Hall effect observed in a tilted beam-detector system, which is believed to result from an effective spin-redirection Berry geometric phase, the geometric spin Hall effect demonstrated here is attributed to an effective, spatially varying Pancharatnam-Berry geometric phase generated by the inhomogeneous polarization geometry. Our further experiments show that the geometric spin Hall effect can be tuned by tailoring the polarization geometry of light, demonstrating the spin states of photons can be steered with a great flexibility.