Extreme Ultraviolet Fractional Orbital Angular Momentum Beams from High Harmonic Generation

Group velocity of light in internal conical refraction

We calculated the group velocity of light in internal conical refraction in a biaxial crystal as a function of the direction of the electric displacement vector, or the vibration direction, of its carrier wave. Our method represents group velocity through the electromagnetic fields of light, rather than its wave normal or ray direction. The travel time of a light pulse traversing a parallel plate biaxial crystal in internal conical refraction is found to vary as a sinusoidal function of twice the vibration angle of the light wave. Our method distinguishes the four directions of the two optic axes in monoclinic and triclinic crystals. Numerical examples are given for K N b O 3 at the wavelength of 400 nm, and for S n 2 P 2 S 6 at the wavelength of 550 nm.

Nonlinear up-conversion of a polarization Möbius strip with half-integer optical angular momentum

Symmetries and conservation laws of energy, linear momentum, and angular momentum play a central role in nonlinear optics. Recently, paraxial light fields with nontrivial topology have been attracting a keen interest. Despite not being eigenstates of the orbital and spin angular momenta (OAM and SAM), they are eigenstates of the generalized angular momentum (GAM) operator-a mixture of the OAM and SAM operators with fractional eigenvalues. By driving high harmonic generation with a polarization Möbius strip carrying a half-integer GAM charge and implementing angular momentum characterization methods in the extreme ultraviolet range, we demonstrate the linear scaling of the GAM with the harmonic order, each harmonic carrying a precise half-integer GAM charge. Our work shows that beyond SAM and OAM, the GAM is, in some situations, an appropriate quantum number. It paves the way for finer manipulations and applications of light beams containing fractional-order polarization singularities.

Role of fractional high harmonics with non-integer OAM on the generation of a helical attosecond pulse train

Extreme-ultraviolet pulses of attosecond duration carrying orbital angular momentum (OAM) can be produced by spectrally filtering vortex high harmonics generated in a gas medium. Here we reveal that fractional high harmonics (FHHs) with non-integer OAM generated by a short duration Laguerre-Gaussian laser beam are origins for the change of helical attosecond pulse train (APT) with azimuthal angle. We show that these harmonics have gap and minimum structures in the annular intensity profile and discontinue phase distribution along azimuthal angle. And each FHH can be expressed as a superposition of OAM modes with integer topological charges. Features of FHH can be identified by coherently combining selected OAM modes. We also uncover that these features are formed after FHH is propagated in gas medium and in vacuum. We finally demonstrate that the generation of FHHs and the dependence of helical APTs on azimuthal angle are changed by varying the macroscopic condition.

Fractional Optical Angular Momentum and Multi-Defect-Mediated Mode Renormalization and Orientation Control in Photonic Crystal Microring Resonators

Whispering gallery modes (WGMs) in circularly symmetric optical microresonators exhibit integer quantized angular momentum numbers due to the boundary condition imposed by the geometry. Here, we show that incorporating a photonic crystal pattern in an integrated microring can result in WGMs with fractional optical angular momentum. By choosing the photonic crystal periodicity to open a photonic band gap with a band-edge momentum lying between that of two WGMs of the unperturbed ring, we observe hybridized WGMs with half-integer quantized angular momentum numbers (m∈Z+1/2). Moreover, we show that these modes with fractional angular momenta exhibit high optical quality factors with good cavity-waveguide coupling and an order of magnitude reduced group velocity. Additionally, by introducing multiple artificial defects, multiple modes can be localized to small volumes within the ring, while the relative orientation of the delocalized band-edge states can be well controlled. Our Letter unveils the renormalization of WGMs by the photonic crystal, demonstrating novel fractional angular momentum states and nontrivial multimode orientation control arising from continuous rotational symmetry breaking. The findings are expected to be useful for sensing and metrology, nonlinear optics, and cavity quantum electrodynamics.

Partially coherent conical refraction promises new counter-intuitive phenomena

In this paper, we extend the paraxial conical refraction model to the case of the partially coherent light using the unified optical coherence theory. We demonstrate the decomposition of conical refraction correlation functions into well-known conical refraction coherent modes for a Gaussian Schell-model source. Assuming randomness of the electrical field phase of the input beam, we reformulated and significantly simplified the rigorous conical refraction theory. This approach allows us to consider the propagation of light through a conical refraction crystal in exactly the same way as in the classical case of coherent radiation. Having this in hand, we derive analytically the conical refraction intensity both in the focal plane and in the far field, which allows us to explain and rigorously justify earlier experimental findings and predict new phenomena. The last include the counterintuitive effect of narrowing of the conical refraction ring width, disappearance of the dark Poggendorff’s ring in the Lloyd’s plane, and shift of Raman spots for the low-coherent conical refraction light. We also demonstrate a universal power-law dependence of conical refraction cones coherence degree on the input correlation length and diffraction-free propagation of the low-coherent conical refraction light in the far field.

Polarization-dependent group velocity of light pulses traveling in the optic ray axis directions of a biaxial crystal

We theoretically prove that the group velocity of a light pulse traveling in an optic ray axis direction of a biaxial crystal depends on the polarization state of the light. Our calculation shows that the group index varies as a sinusoidal function of twice the polarization angle of the light pulse. For monoclinic and triclinic crystals, in general the four directions of the two optic ray axes need to be distinguished. Numerical examples show that in KNbO₃ the group velocity varies by 2.7% at 400 nm wavelength, and in Sn₂P₂S₆ it varies by 3.9% at 550 nm wavelength, when the polarization state of the light is changing.

Conservation laws for electron vortices in strong-field ionisation

ABSTRACT

We investigate twisted electrons with a well-defined orbital angular momentum, which have been ionised via a strong laser field. By formulating a new variant of the well-known strong field approximation, we are able to derive conservation laws for the angular momenta of twisted electrons in the cases of linear and circularly polarised fields. In the case of linear fields, we demonstrate that the orbital angular momentum of the twisted electron is determined by the magnetic quantum number of the initial bound state. The condition for the circular field can be related to the famous ATI peaks, and provides a new interpretation for this fundamental feature of photoelectron spectra. We find the length of the circular pulse to be a vital factor in this selection rule and, employing an effective frequency, we show that the photoelectron OAM emission spectra are sensitive to the parity of the number of laser cycles. This work provides the basic theoretical framework with which to understand the OAM of a photoelectron undergoing strong field ionisation.

Close relationship between Bessel-Gaussian and conical refraction beams

We demonstrate that the conical refraction of the input elegant Laguerre-Gaussian beams can be effectively described through generalized Bessel-Gaussian light beams. We performed numerical simulations and show good agreement between the exact solution and our proposed Bessel-Gaussian approximation model. Physical clarity of the proposed model has allowed us to explain the transition of the classical double-ring pattern of conical refraction in the Lloyd plane into a multi-ring one and predict new phenomenon such as the Raman spot shift and dependence of the conical refraction ring radius on the value of the orbital angular momentum.

Dynamic spatiotemporal beams that combine two independent and controllable orbital-angular-momenta using multiple optical-frequency-comb lines

Novel forms of beam generation and propagation based on orbital angular momentum (OAM) have recently gained significant interest. In terms of changes in time, OAM can be manifest at a given distance in different forms, including: (1) a Gaussian-like beam dot that revolves around a central axis, and (2) a Laguerre-Gaussian (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$LG_{\ell ,p}$$\end{document}LGℓ,p) beam with a helical phasefront rotating around its own beam center. Here we explore the generation of dynamic spatiotemporal beams that combine these two forms of orbital-angular-momenta by coherently adding multiple frequency comb lines. Each line carries a superposition of multiple \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$LG_{\ell ,p}$$\end{document}LGℓ,p modes such that each line is composed of a different \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell$$\end{document}ℓ value and multiple p values. We simulate the generated beams and find that the following can be achieved: (a) mode purity up to 99%, and (b) control of the helical phasefront from 2π-6π and the revolving speed from 0.2–0.6 THz. This approach might be useful for generating spatiotemporal beams with even more sophisticated dynamic properties.

Orbital angular momentum takes several forms in structured light beams. Here, the authors demonstrate control of dynamic spatiotemporal beams combining two forms of orbital angular momenta, by coherently adding frequency comb lines.

Effects of the Coupling between the Orbital Angular Momentum and the Temporal Degrees of Freedom in the Most Intense Ring of Ultrafast Vortices

It has recently been shown that the temporal and the orbital angular momentum (OAM) degrees of freedom in ultrafast (few-cycle) vortices are coupled. This coupling manifests itself with different effects in different parts of the vortex, as has been shown for the ring surrounding the vortex where the pulse energy is maximum, and also in the immediate vicinity of the vortex center. However, in many applications, the ring of maximum energy is not of primary interest, but the one where the peak intensity of the pulse is maximum, which is particularly true in nonlinear optics applications such as experiments with ultrafast vortices that excite high harmonics and attosecond pulses that also carry OAM. In this paper, the effects of the OAM-temporal coupling on the ring of maximum pulse peak intensity, which do not always coincide with the ring of maximum pulse energy, are described. We find that there is an upper limit to the magnitude of the topological charge that an ultrafast vortex with a prescribed pulse shape in its most intense ring can carry, and vice versa, a lower limit to the pulse duration in the most intense ring for a given magnitude of the topological charge. These limits imply that, with a given laser source spectrum, the duration of the synthesized ultrafast vortex increases with the magnitude of the topological charge. Explicit analytical expressions are given for the ultrafast vortices that contain these OAM-temporal couplings effects, which may be of interest in various applications, in particular in the study of their propagation and interaction with matter.

Fractional vortex ultrashort pulsed beams with modulating vortex strength

In most papers about the fractional vortex continuous beams (FVCBs), the relationship between the total vortex strength S_α and the propagation distance is not analyzed since the vortex structure is not stable in the near field. In this paper, we theoretically study the fractional vortex ultrashort pulsed beams (FVUPBs) possessing non-integer topological charges α at arbitrary plane and find that the vortex structure is propagation-distance-dependent. Both the intensity and phase distributions are calculated to analyze the vortex structure. To evaluate the propagation properties of FVUPBs, we focus on the total vortex strength (TVS) of FVUPBs to investigate the number of vortex, and demonstrate that the birth of a vortex is at α = m + ɛ, where m is an integer, ɛ is a changing fraction depending on the pulse durations, peak wavelengths and propagation distances. Furthermore, we discover that the FVUPBs carry decreasing TVS along the propagation axis in free space. This special vortex structure for FVUPBs appears due to the mixture weight of vortex pulsed beam with different integer topological charges (TCs) n. However, the total orbital angular momentum is invariant during propagation. The above phenomenon presented in our paper are totally particular and intriguing compared with the FVCBs.

Generation of light beams with custom orbital angular momentum and tunable transverse intensity symmetries

We introduce a novel and simple modulation technique to tailor optical beams with a customized amount of orbital angular momentum (OAM). The technique is based on the modulation of the angular spectrum of a seed beam, which allows us to specify in an independent manner the value of OAM and the shape of the resulting beam transverse intensity. We experimentally demonstrate our method by arbitrarily shaping the radial and angular intensity distributions of Bessel and Laguerre-Gauss beams, while their OAM value remains constant. Our experimental results agree with the numerical and theoretical predictions.

Generation of extreme-ultraviolet beams with time-varying orbital angular momentum

Light fields carrying orbital angular momentum (OAM) provide powerful capabilities for applications in optical communications, microscopy, quantum optics, and microparticle manipulation. We introduce a property of light beams, manifested as a temporal OAM variation along a pulse: the self-torque of light. Although self-torque is found in diverse physical systems (i.e., electrodynamics and general relativity), it was not realized that light could possess such a property. We demonstrate that extreme-ultraviolet self-torqued beams arise in high-harmonic generation driven by time-delayed pulses with different OAM. We monitor the self-torque of extreme-ultraviolet beams through their azimuthal frequency chirp. This class of dynamic-OAM beams provides the ability for controlling magnetic, topological, and quantum excitations and for manipulating molecules and nanostructures on their natural time and length scales.

Conservation of Torus-knot Angular Momentum in High-order Harmonic Generation

High-order harmonic generation stands as a unique nonlinear optical up-conversion process, mediated by a laser-driven electron recollision mechanism, which has been shown to conserve energy, linear momentum, and spin and orbital angular momentum. Here, we present theoretical simulations that demonstrate that this process also conserves a mixture of the latter, the torus-knot angular momentum J_{γ}, by producing high-order harmonics with driving pulses that are invariant under coordinated rotations. We demonstrate that the charge J_{γ} of the emitted harmonics scales linearly with the harmonic order, and that this conservation law is imprinted onto the polarization distribution of the emitted spiral of attosecond pulses. We also demonstrate how the nonperturbative physics of high-order harmonic generation affect the torus-knot angular momentum of the harmonics, and we show that this configuration harnesses the spin selection rules to channel the full yield of each harmonic into a single mode of controllable orbital angular momentum.

Application of optimized waveforms for enhancing high-harmonic yields in a three-color laser-field synthesizer

We apply the optimization method suggested by Jin et al. [Nat. Commun.5, 4003 (2014)24873949] to a three-color laser-field synthesizer in a recent experiment by Burger et al. [Opt. Express25(25), 31130 (2017)29245790] for efficient high-order harmonic generation (HHG). With the experimental laser parameters being precisely tuned according to those returned by the genetic optimization, the three-color waveform composed by a 790-nm laser with its second and third harmonic fields, can enhance the macroscopic HHG yields by one to two orders with only 80% pulse energy compared to the fundamental single-color field. We check that this enhancement can be realized for He or Ne gas at both low and high gas pressures. The optimized waveform enables the short-trajectory emissions dominant to facilitate the buildup of the harmonic field, which is revealed by analyzing the behaviors of electron trajectories and the time-frequency pictures of the single-atom and macroscopic HHG. We also optimize the two-color waveform consisting of the fundamental laser and its third harmonic field for the flexible choice in the experiment. This study provides with a practical route to implement the optimization technique in the experiment for the high-flux harmonic generation from the extreme ultraviolet to the X-rays.

Utilizing the Momentum in Orbital Angular Momentum: Augmented OAM induced by a [Formula: see text] Aperture of Three Elements

The feasibility to induce augmented dominant OAM modes by a π/2 aperture of three elements in space and weighted quasi-phase shifts is realised in this paper. It is shown through theory, numerical simulations and experimentation, that electromagnetic (EM) waves carrying non-integer OAM with dominant mode l = +1 in the microwave domain can be generated by a quarter of a full azimuthal annular aperture consisting of three elements and a weighted phase shift augmenting the expected conventional phase shift to reach Berry's mode dominance theory of half integer l. With reference to the uncertainty principle of angular momentum and angular position, the proposed augmented OAM with weighted phase shift method seems to decrease mode uncertainties and augment mode dominance.

Conservation of orbital angular momentum for high harmonic generation of fractional vortex beams

This work demonstrates conservation of average orbital angular momentum for high harmonic generation of fractional vortex beams. High harmonics are generated in reflected light beams in a three-dimensional particle-in-cell simulation. The average orbital angular momentum of the beam is calculated when a relativistic linearly polarized fractional vortex beam impinges on a solid foil. The harmonic generation progress can be well explained by using the vortex oscillating mirror model. Both simulation and theoretical analysis show that the average orbital momentum of the nth harmonic is n times that of the fundamental frequency beam. This provides evidence that the average orbital angular momentum obeys momentum conservation during the harmonic generation of fractional vortex beams.

Controlling the multi-electron dynamics in the high harmonic spectrum from N2O molecule using TDDFT

In this study, high harmonic generation from a multi-atomic nitrous oxide molecule was investigated. A comprehensive three-dimensional calculation of the molecular dynamics and electron trajectories through an accurate time-dependent density functional theory was conducted to efficiently explore a broad harmonic plateau. The effects of multi-electron and inner orbitals on the harmonic spectrum and generated coherent attosecond pulses were analyzed. The role of the valence electrons in controlling the process and extending the harmonic plateau was investigated. The main issue of producing a super-continuum harmonic spectrum via a frequency shift was considered. The time-frequency representation by means of a wavelet transform of the induced dipole acceleration provided a good insight into the distorted effects from the nonlinear processes in high harmonic emission. The effect of the chirped laser pulse on the production of broadband amplitude was justified in this model. By adjusting the optimal laser parameters to an input intensity of 2.5 × 10¹⁴ W cm^-2, an isolated 68 as pulse was generated.

Controlling the polarization and vortex charge of attosecond high-harmonic beams via simultaneous spin-orbit momentum conservation

Optical interactions are governed by both spin and angular momentum conservation laws, which serve as a tool for controlling light-matter interactions or elucidating electron dynamics and structure of complex systems. Here, we uncover a form of simultaneous spin and orbital angular momentum conservation and show, theoretically and experimentally, that this phenomenon allows for unprecedented control over the divergence and polarization of extreme-ultraviolet vortex beams. High harmonics with spin and orbital angular momenta are produced, opening a novel regime of angular momentum conservation that allows for manipulation of the polarization of attosecond pulses-from linear to circular-and for the generation of circularly polarized vortices with tailored orbital angular momentum, including harmonic vortices with the same topological charge as the driving laser beam. Our work paves the way to ultrafast studies of chiral systems using high-harmonic beams with designer spin and orbital angular momentum.