Concept Of Geometric Phase Research Articles

On the role of geometric phase in the dynamics of elastic waveguides.

The geometric phase provides important mathematical insights to understand the fundamental nature and evolution of the dynamic response in a wide spectrum of systems ranging from quantum to classical mechanics. While the concept of geometric phase, which is an additional phase factor occurring in dynamical systems, holds the same meaning across different fields of application, its use and interpretation can acquire important nuances specific to the system of interest. In recent years, the development of quantum topological materials and its extension to classical mechanical systems have renewed the interest in the concept of geometric phase. This review revisits the concept of geometric phase and discusses, by means of either established or original results, its critical role in the design and dynamic behaviour of elastic waveguides. Concepts of differential geometry and topology are put forward to provide a theoretical understanding of the geometric phase and its connection to the physical properties of the system. Then, the concept of geometric phase is applied to different types of elastic waveguides to explain how either topologically trivial or non-trivial behaviour can emerge based on the geometric features of the waveguide. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 2)'.

Recent Advances in Geometric Phase Metasurfaces: Principles and Applications

AbstractThe concept of geometric phase traversing numerous domains in physics and has been a continuous source of fascination and inspiration for researchers. Despite the extensive research surrounding geometric phase from decades, advances in technology continue to yield novel theories, innovative devices, and captivating applications, extending even to the realm of subwavelength scales. This review article provides a comprehensive exploration of geometric phase metasurfaces, delving into their design principles and categorizing them based on materials properties. In addition, multi‐fold and reconfigurable metasurfaces based on geometric principle are further explored with their unique capabilities and potential impact on a diverse range of applications, including beam steering, lensing, polarization conversion, and holographic imaging. By examining the state‐of‐the‐art in geometric phase metasurfaces, insights are aimed to offer into their current capabilities and limitations. Finally, the prospects and challenges are discussed that lie ahead for this promising field, paving the way for future advancements and innovations.

Full-Dimensional Geometric-Phase Spatial Light Metamodulation.

Full-dimensional spatial light modulation requires simultaneous, arbitrary, and independent manipulation of the spatial phase, amplitude, and polarization. This is crucial for leveraging the complete physical dimension resources of light. However, full-dimensional metamodulation can be challenging due to the need for multiple independent control factors. To address this challenge, here we propose parallel-tasking metasurfaces to enable full-dimensional spatial light metamodulation based fully on the geometric-phase concept. Indeed, the meta-atoms are divided into several subphases, each of which serves as an independent control factor to manipulate light phase, amplitude, and polarization through geometric phase, interference, and orthogonal polarization superposition, respectively. Therefore, the macroscopic group of meta-atoms leads to metasurfaces that can achieve broadband full-dimensional spatial light metamodulation, as demonstrated by various types of structured light generation. This approach paves the way to future wide applications of light manipulation enabled by full-dimensional spatial light metamodulation.

High‐Efficiency Dual‐Band Metasurface with Independent Multifold Geometric Phases

AbstractThe concept of geometric phase has attracted enormous interest in the metasurface community for achieving phase variation with simple geometrical rotation. Here, this work demonstrates for the first time a highly efficient dual‐band metasurface with independent phase responses for structures with different‐fold rotational symmetries. For proof of concept, a hybrid unit cell structure with nested resonators of different‐fold (single‐fold and threefold) is proposed, where geometric rotation independent phase shifts of two times and six times the rotation angle are achieved in lower and higher bands, respectively. By rotating the resonators within the unit cell, highly efficient independent full phase controls can be achieved with circularly polarized incident waves. The co‐polarized reflection efficiency exceeds 90% at the frequency domain from 10.95 to 11.75 GHz (corresponding to the inner resonator) and exceeds 80% between 14.10 and 14.43 GHz (corresponding to the outer resonator), with peak values of 94.7% at 11.55 GHz and 85.5% at 14.23 GHz, respectively. To validate the versatility of the design paradigm, a metasurface porotype is fabricated and experimentally tested by generating independent holographic images.

Geometric Phase Control of Surface Plasmons by Dipole Sources

AbstractGeometric phase metasurfaces, as one of the main branches of meta‐optics, have attracted enormous interest in the last two decades. Recently, through rotating a set of subwavelength dipole sources, geometric phase concept has been extended to near‐field regime for the control of surface plasmons (SPs). Despite this progress, puzzles and shortcomings still exist: it is curious that geometric phases equal to once and twice the rotation angle of dipole source are both reported for SP controls, and the control strategies examined thus far only work for a single wavelength. Hereby, a rigorous derivation of the SP excitation of dipole sources upon circularly polarized illumination is performed, and the rotation dependence and in‐plane coordinate correlation of geometric phase control of SPs is clarified. Moreover, a holographic approach is proposed to implement multiplexed geometric phase control, experimentally demonstrating several metalenses that can couple and steer the incident circular polarizations of four wavelengths and two spin directions to different SP focusing beams. This work will pave an avenue toward the development of integrated and multiplexed SP devices.

Liquid crystalʼs photoalignment for formation of phase profiles via geometric phase distribution

Basic understandings on the concept of geometric phase, also known as Pancharatnam – Berry phase, and its application to creation of photonic liquid crystal devices as thin-films of photoaligned nematic liquid crystals are presented. The significance of the strong azimuthal anchoring energy and the role of birefringence in liquid crystal photoalignment layer for formation of geometric phase gradients are shown. The dependence of phase retardation of circular polarised light passing through the half-wave phase plate on the azimuthal angle of the plate orientation is explained in details, as it gives ground to formation of geometric phase distribution of optical liquid crystal devices, working in circular polarised light. The effective refractive index is introduced for characterisation of the optical properties of linear periodic liquid crystal’s structure that forms profile of geometric phase surface. The successful implementations of photonic liquid crystal devices (polarisation diffraction grating, annular diffraction grating, q-plate, q-plate with a phase core) are analysed in terms of geometric phase distributions and the corresponding equations of profile of the phase surfaces that ensure functioning of the devices are presented.

Classical and generalized geometric phase in electromagnetic metasurfaces

The geometric phase concept has profound implications in many branches of physics, from condensed matter physics to quantum systems. Although geometric phase has a long research history, novel theories, devices, and applications are constantly emerging with developments going down to the subwavelength scale. Specifically, as one of the main approaches to implement gradient phase modulation along a thin interface, geometric phase metasurfaces composed of spatially rotated subwavelength artificial structures have been utilized to construct various thin and planar meta-devices. In this paper, we first give a simple overview of the development of geometric phase in optics. Then, we focus on recent advances in continuously shaped geometric phase metasurfaces, geometric–dynamic composite phase metasurfaces, and nonlinear and high-order linear Pancharatnam–Berry phase metasurfaces. Finally, conclusions and outlooks for future developments are presented.

Multi-beam multi-mode vortex beams generation based on metasurface in terahertz band

The generating of vortex beams in the terahertz (THz) band attracts significant attention due to their applications in high-speed communication and high-resolution imaging. In this article, a novel reflective metasurface working in the THz band is designed to generate four vortex beams with different topological charges in different directions. The unit cell is designed based on the geometric phase, and it consists of two metallic (gold) layers and one dielectric layer in between. The top layer of the unit cell includes an elliptic patch and a circular ring, and the bottom layer of the unit cell is a metallic ground. The reflection efficiency of the unit cell is very high due to the presence of metallic ground. To break through the limitations of traditional methods, the metasurface is a good choice to generate beams that carry orbital angular momentum (OAM). Using the concept of geometric phase, the reflection phase of reflective circular polarization (CP) electromagnetic waves can be controlled in an ingenious way. Owing to the property of the geometric phase, inverse phase shift can be achieved for left-handed circular polarization and right-handed circular polarization waves. By utilizing this trait of geometric phase, one can decompose a linear polarization wave into two orthogonal circular polarization waves and control their properties respectively. By rotating the top layer of the unit cell, 360-degree phase shift and the phase distribution satisfying the requirement for generating the multi-beam multi-mode vortex beam can be achieved. In order to control the direction and the topological charge of each beam, based on the geometric phase, the theory of reflectarray and the phase composition principle, the phase distribution of the reflective metasurface is calculated to provide the phase compensation to make the vortex beams point to certain directions. It is worthwhile to point out that the method presented in this paper provides a way to generate complex multi-mode vortex beams in the THz band. The simulations and measurements show that the metasurface can generate four vortex beams with topological charges <i>l</i> = ± 1 and ± 2 in different directions in the THz band. These results also indicate that our design has great potential applications in wireless communication and high-resolution imaging.

Geometric phase for “dark” subspaces in coherent population trapping

The concept of geometric phase acquired by a quantum state during its evolution is generalized to the subspace of states of evolving quantum system. Physical motivation of this generalization comes from the effect of coherent population trapping. Under certain conditions, there exists a two-dimensional “dark” subspace of atomic ground states that does not interact with the external radiation. The structure of this “dark” subspace depends on the local field amplitude. We show that certain mathematical difficulties in finding the geometric phase for “dark” subspaces can be circumvented with the help of the orthogonal “bright” subspace.

Polarization multiplexing and bifocal optical vortex metalens

In this article, we demonstrate several optical vortex metalens (OVMs) based on the Pancharatnam–Berry (PB) geometric phase concept utilizing silicon nanobricks. The nanostructured OVMs are designed via coalescing the phase profiles of metalens and vortex phase plate, which can produce and focus the vortex beam simultaneously. Through analyzing four single-focal OVMs suitable for left circularly polarized (LCP) or right circularly polarized (RCP) incident light respectively, we find that the topological sign of vortex light is strongly associated with polarization state of incident light. Through multiplexing two OVMs only suitable for one kind of circularly polarized light, we design one polarization multiplexing OVM, which can focus different circularly polarized (CP) vortex beam to the same position. We also design one longitudinal bifocal OVM, which can focus LCP vortex beam to the different positions. Thus, these OVMs have the promising applications in integrated optics, classical and quantum optics, and vortex optics.

Analysis of the geometric phase for a nanowire-bridged superconducting Fabry-Perot resonator

The geometric phases of a nanowire-bridged superconducting Fabry-Perot resonator subjected to a microwave transmission have been investigated through its modelling into a RLC-circuit. Because the Hamiltonian of the system is a somewhat complicated form, special mathematical techniques, such as the invariant operator method and the unitary transformation approach, have been adopted in order to treat the system; These methods are very useful for managing complicated time-dependent Hamiltonian systems. We have rigorously evaluated the analytical geometric phases in both the Fock and coherent states. Typically, the geometric phases oscillate and the amplitude of such oscillations tend to grow over time. The influence of parameters of the system on the geometric phases has been analyzed in detail through the relevant illustrations. From our research, the concept of geometric phases and associated quantum mechanical characters of the system has been clarified. Our investigation for the geometric phases is useful for understanding topological features of the system, that take place through the evolution of the wave functions.

Thermal Uhlmann-Chern number from the Uhlmann connection for extracting topological properties of mixed states

The Berry phase is a geometric phase of a pure state when the system is adiabatically transported along a loop in its parameter space. The concept of geometric phase has been generalized to mixed states by the so called Uhlmann phase. However, the Uhlmann phase is constructed from the Uhlmann connection that possesses a well defined global section. This property implies that the Uhlmann connection is topologically trivial and as a consequence, the corresponding Chern character vanishes. We propose modified Chern character whose integral gives the thermal Uhlmann Chern number, which is related to the winding number of the mapping defined by the Hamiltonian. Therefore, the thermal Uhlmann Chern number reflects the topological properties of the underlying Hamiltonian of a mixed state. By including the temperature dependence in the volume integral, we also introduce the non-topological thermal Uhlmann Chern number which varies with temperature but is not quantized at finite temperatures. We illustrate the applications to a two-band model and a degenerate four-band model.

STEM moiré analysis for 2D strain measurements.

A moiré pattern is created in a scanning transmission electron microscope (STEM) when the scan step is close to a crystalline periodicity. Usually, fringes are visible in only one direction, corresponding to a single set of lattice planes, but fringes can be formed in two directions or more. Using an accurate independent calibration, the strains in silicon devices have been determined from the spacing and orientation of one-directional STEM moiré fringes. In this report, we first discuss the origin of the STEM moiré, and then we show how an accurate calibration of the scan step can be obtained from the STEM moiré pattern itself, providing that we know initially only an approximate scan step and the planar spacing. The new calibration scheme also makes the STEM moiré experiments easier, since it can be applied for the moiré where the scan direction is not precisely aligned with the crystalline lattice. Finally, we show how the two-dimensional strain information will be readily extracted from two one-directional moiré patterns using the concept of geometric phase.

Geometric phase and topology of elastic oscillations and vibrations in model systems: Harmonic oscillator and superlattice

We illustrate the concept of geometric phase in the case of two prototypical elastic systems, namely the one-dimensional harmonic oscillator and a one-dimensional binary superlattice. We demonstrate formally the relationship between the variation of the geometric phase in the spectral and wave number domains and the parallel transport of a vector field along paths on curved manifolds possessing helicoidal twists which exhibit non-conventional topology.

Geometric phases in discrete dynamical systems

In order to study the behaviour of discrete dynamical systems under adiabatic cyclic variations of their parameters, we consider discrete versions of adiabatically-rotated rotators. Parallelling the studies in continuous systems, we generalize the concept of geometric phase to discrete dynamics and investigate its presence in these rotators. For the rotated sine circle map, we demonstrate an analytical relationship between the geometric phase and the rotation number of the system. For the discrete version of the rotated rotator considered by Berry, the rotated standard map, we further explore this connection as well as the role of the geometric phase at the onset of chaos. Further into the chaotic regime, we show that the geometric phase is also related to the diffusive behaviour of the dynamical variables and the Lyapunov exponent.

All-silicon nanorod-based Dammann gratings.

Established diffractive optical elements (DOEs), such as Dammann gratings, whose phase profile is controlled by etching different depths into a transparent dielectric substrate, suffer from a contradiction between the complexity of fabrication procedures and the performance of such gratings. In this Letter, we combine the concept of geometric phase and phase modulation in depth, and prove by theoretical analysis and numerical simulation that nanorod arrays etched on a silicon substrate have a characteristic of strong polarization conversion between two circularly polarized states and can act as a highly efficient half-wave plate. More importantly, only by changing the orientation angles of each nanorod can the arrays control the phase of a circularly polarized light, cell by cell. With the above principle, we report the realization of nanorod-based Dammann gratings reaching diffraction efficiencies of 50%-52% in the C-band fiber telecommunications window (1530-1565 nm). In this design, uniform 4×4 spot arrays with an extending angle of 59°×59° can be obtained in the far field. Because of these advantages of the single-step fabrication procedure, accurate phase controlling, and strong polarization conversion, nanorod-based Dammann gratings could be utilized for various practical applications in a range of fields.

Malus Law in terms of the geometric phase

The aim of the present work is the description of the commonly known Malus law as the interference of two waves. We have proposed two cases: interference of two waves with different intensities and equal phases and interference of two waves with different phases and equal intensities. The first case can be realized by decomposition of the polarization state of the light incident on the analyzer into base of linearly polarized waves. In the second case the base of circularly polarized waves should be chosen. This last option allows applying of the geometric phase concept to the Malus law. Full Text: PDF References P. Kurzynowski, W.A. Woźniak, M. Szarycz, "Geometric phase: two triangles on the Poincaré sphere", J. Opt. Soc. Am. A 28, pp. 475-482 (2011). CrossRef W.A. Woźniak, P. Kurzynowski, M. Zdunek, "Malus Law – interferometric interpretation", Opt. Appl. Vol. XLIII, No. 2, (2013). DirectLink

Experimental realization of non-Abelian non-adiabatic geometric gates

The geometric aspects of quantum mechanics are emphasized most prominently by the concept of geometric phases, which are acquired whenever a quantum system evolves along a path in Hilbert space, that is, the space of quantum states of the system. The geometric phase is determined only by the shape of this path and is, in its simplest form, a real number. However, if the system has degenerate energy levels, then matrix-valued geometric state transformations, known as non-Abelian holonomies--the effect of which depends on the order of two consecutive paths--can be obtained. They are important, for example, for the creation of synthetic gauge fields in cold atomic gases or the description of non-Abelian anyon statistics. Moreover, there are proposals to exploit non-Abelian holonomic gates for the purposes of noise-resilient quantum computation. In contrast to Abelian geometric operations, non-Abelian ones have been observed only in nuclear quadrupole resonance experiments with a large number of spins, and without full characterization of the geometric process and its non-commutative nature. Here we realize non-Abelian non-adiabatic holonomic quantum operations on a single, superconducting, artificial three-level atom by applying a well-controlled, two-tone microwave drive. Using quantum process tomography, we determine fidelities of the resulting non-commuting gates that exceed 95 per cent. We show that two different quantum gates, originating from two distinct paths in Hilbert space, yield non-equivalent transformations when applied in different orders. This provides evidence for the non-Abelian character of the implemented holonomic quantum operations. In combination with a non-trivial two-quantum-bit gate, our method suggests a way to universal holonomic quantum computing.

Adiabatic geometric phase in the nonlinear coherent coupler

We introduce the concept of geometric phase to the nonlinear coherent coupler. With considering the adiabatic change of the distance-dependent phase mismatch, we calculate the adiabatic geometric phase related to the supermode of the coupler analytically. We find that the phase depends on the input light intensity explicitly. In particular, in the low and high intensity limits, the phase equals half of the area on the Poincare sphere enclosed by the evolution loop of the system. At the critical intensity where different supermodes merge, the phase diverges, which can be considered as the signal of a continuous phase transition.

A new kind of geometric phases in open quantum systems and higher gauge theory

A new approach is proposed, extending the concept of geometric phases to adiabatic open quantum systems described by density matrices (mixed states). This new approach is based on an analogy between open quantum systems and dissipative quantum systems which uses a C*-module structure. The gauge theory associated with these new geometric phases does not employ the usual principal bundle structure but a higher structure, a categorical principal bundle (the so-called principal 2-bundle or non-Abelian bundle gerbe) which is sometimes a non-Abelian twisted bundle. The need to site the gauge theory in this higher structure is a geometrical manifestation of the decoherence induced by the environment on the quantum system.