Abstract

Vortices are whirling disturbances commonly found in nature ranging from tremendously small scales in Bose-Einstein condensates to cosmologically colossal scales in spiral galaxies. An optical vortex, generally associated with a spiral phase, can carry orbital angular momentum (OAM). The optical OAM can either be in the longitudinal direction if the spiral phase twists in the spatial domain or in the transverse direction if the phase rotates in the spatiotemporal domain. In this article, we demonstrate the intersection of spatiotemporal vortices and spatial vortices in a wave packet. As a result of this intersection, the wave packet hosts a tilted OAM that provides an additional degree of freedom to the applications that harness the OAM of photons.

Keywords: orbital angular momentum, spatiotemporal vortex, spatial vortex, spiral phase

Introduction

Vortices, ubiquitous in nature, are circulating disturbances of liquid, gas or other media. They have been found in turbulent water, circulating air around wingtips, swirling galaxies and optics as well [1]. Optical vortices are generally associated with a spiral wavefront with phase singularities of zero intensity. The twisted wavefront gives rise to an azimuthal component of Poynting vector that contributes to an integrated orbital angular momentum (OAM) pointing along the beam axis. Each photon carries an OAM of 𝑙ℏ where ℏ is the reduced Planck's constant and l is an integer, generally referred to as the topological charge [2]. The connection of vortex beams with optical OAM has spurred substantial theoretical and experimental research and found a wealth of applications in both classical and quantum optics [3-10].

Recent theoretical studies manifest that optical OAM does not have to be longitudinal but can be tilted to the optical axis [11,12]. The tilted OAM could be realized with a fast-moving observer close to the speed of light. Experimental progress has shown that a small fraction of optical energy can circulate in a spatiotemporal plane in a nonlinear interaction of an extremely high-power laser pulse and air [13]. Contrary to the longitudinal OAM that is associated with a spiral phase in the spatial domain, the transverse OAM roots in a spiral phase in the spatiotemporal domain that rotates around an axis perpendicular to the propagation direction. Although experimentally explored, it is still a challenging task to control and manipulate a spiral phase with circulating Poynting vector in a spatiotemporal plane in a linear manner. The difficulty has recently been overcome by forming a spiral phase in the spatial frequency-temporal frequency domain and retaining the spiral phase in the spatiotemporal domain through a two-dimensional spatiotemporal Fourier transform [14-16].

The intersection of spatial vortices has been reported in literature [17]. However, the interacting dynamics stays at the intersection point and does not travel with the beam. In this work, we experimentally demonstrate the intersection of spatiotemporal vortices and spatial vortices in an optical wave packet. The wave packet contains both screw and edge dislocations in phase. The intersection of two distinct types of optical vortices reveals interesting three-dimensional energy flow that travels at the speed of light. The combination of transverse OAM carried by spatiotemporal vortices and longitudinal OAM carried by spatial vortices gives rise to a tilted OAM with respect to the optical axis. The average three-dimensional OAM per photon remain unchanged after propagation in free space. The tilted OAM is fully controllable in value and orientation through the topological charges of the two types of vortices.

Theoretical Background

Fundamentals of Optical Vortices

Optical vortices represent phase singularities in electromagnetic waves where the phase becomes undefined and the intensity drops to zero. These singularities are characterized by their topological charge, which determines the number of 2π phase cycles around the singularity. The mathematical description of an optical vortex beam typically involves Laguerre-Gaussian modes, which contain a spiral phase term exp(ilφ), where l is the topological charge and φ is the azimuthal angle.

Orbital Angular Momentum in Photonics

The orbital angular momentum (OAM) of light arises from the helical phase structure of optical vortices. Each photon in an OAM-carrying beam possesses an angular momentum of lℏ, where l is the topological charge. This OAM is distinct from the spin angular momentum associated with circular polarization. The Poynting vector in such beams follows a spiral trajectory, resulting in the characteristic orbital angular momentum.

Spatiotemporal Vortices

Spatiotemporal vortices represent a more recent development in vortex physics, where the phase singularity exists not only in space but also evolves in time. These vortices are characterized by their ability to carry transverse OAM, meaning the angular momentum vector is perpendicular to the propagation direction. The generation of spatiotemporal vortices typically involves precise manipulation of both spatial and temporal degrees of freedom in optical pulses.

Experimental Methodology

Wave Packet Generation

The experimental setup for generating wave packets containing both spatial and spatiotemporal vortices involves several key components. A mode-locked laser source provides the initial pulses, which are then spatially and temporally shaped using spatial light modulators (SLMs) and pulse shapers. The precise control over both spatial and temporal phases enables the creation of the desired vortex structures.

Intersection Technique

The intersection of spatial and spatiotemporal vortices is achieved through careful design of the phase profiles in both domains. By implementing specific phase masks that combine spiral phases in spatial coordinates with appropriate temporal phase modulations, we can create wave packets where both types of vortices coexist and interact.

Measurement and Characterization

The characterization of the resulting tilted OAM involves several diagnostic techniques. Interferometric methods are used to visualize the phase structure, while momentum measurements provide quantitative information about the OAM components. The three-dimensional energy flow is mapped using specialized detectors capable of resolving both spatial and temporal variations.

Results and Analysis

Tilted OAM Generation

Our experiments successfully demonstrate the generation of tilted orbital angular momentum through the intersection of spatial and spatiotemporal vortices. The resulting OAM vector is oriented at an angle to the optical axis, with the specific angle determined by the relative strengths of the longitudinal and transverse OAM components.

Three-Dimensional Energy Flow

The intersection of vortices creates complex three-dimensional energy flow patterns that propagate at the speed of light. These patterns exhibit both circulating and translating components, representing the combined effects of the spatial and spatiotemporal vortices.

Control Parameters

The value and orientation of the tilted OAM are fully controllable through manipulation of the topological charges of the constituent vortices. By independently varying the spatial and spatiotemporal topological charges, we can achieve precise control over both the magnitude and direction of the resulting OAM vector.

Applications

Optical Spanners and Manipulation

The ability to create tilted OAM opens new possibilities for optical manipulation techniques. Optical spanners with arbitrary three-dimensional orientation can be created, providing controllable torque for rotating microscopic particles in multiple directions simultaneously.

Enhanced Optical Tweezing

The additional degree of freedom provided by tilted OAM enhances the capabilities of optical tweezers, allowing more complex manipulation of particles and biological specimens. The three-dimensional control over rotational forces enables new applications in biophysics and materials science.

Quantum Communications

In quantum information processing, the controllable orientation of OAM provides additional encoding dimensions for quantum states. This could lead to increased channel capacity and enhanced security in quantum communication systems.

Spin-Orbit Angular Momentum Coupling

The tilted OAM platform enables new studies of spin-orbit interactions in photonics. The controlled orientation of OAM relative to spin angular momentum facilitates investigations of fundamental light-matter interactions and may lead to new photonic devices.

Key Insights

• The intersection of spatial and spatiotemporal vortices enables the generation of tilted orbital angular momentum
• The tilted OAM provides an additional degree of freedom for optical manipulation and communication applications
• The three-dimensional energy flow associated with tilted OAM propagates at light speed and maintains stability during propagation
• Precise control over OAM orientation is achievable through manipulation of topological charges
• This platform bridges the gap between purely spatial and purely temporal vortex phenomena

Conclusion

We have demonstrated the experimental realization of tilted orbital angular momentum through the intersection of spatiotemporal and spatial vortices in optical wave packets. This approach provides full control over both the value and orientation of the photonic OAM, creating an additional degree of freedom for applications harnessing optical angular momentum. The tilted OAM maintains its properties during propagation in free space and offers new capabilities for optical manipulation, quantum communications, and fundamental studies of light-matter interactions. The platform developed in this work opens exciting possibilities for three-dimensional control of optical forces and torques, with potential applications across multiple fields including optical trapping, quantum optics, and photonic device engineering.

References

1. Reference 1: Fundamental studies of optical vortices
2. Reference 2: Allen et al., Orbital angular momentum of light
3. Reference 3-10: Various applications of OAM in classical and quantum optics
4. Reference 11-12: Theoretical studies of tilted OAM
5. Reference 13: Experimental progress in spatiotemporal vortices
6. Reference 14-16: Spatiotemporal Fourier transform techniques
7. Reference 17: Intersection of spatial vortices