A generalized K-space coherent averaging method for engineering lattices of spin-orbit beams

Abstract

Spin-orbit beams, in which the orbital angular momentum degree of freedom is coupled to a two-level system such as polarization of light or spin in electrons and neutrons, have gained significant interest for their unique propagation properties and potential applications in imaging, material characterization, optical trapping, and quantum information processing. In this work we introduce a method for generating and engineering two-dimensional lattices of such spin-orbit beams based on coherent averaging in k-space. By programming the angle, amplitude, and polarization of a set of input beams we obtain precise control over lattice geometry and period, as well as the orbital and radial degrees of freedom inside each unit cell. We explore both electromagnetic and matter wave implementations, and we experimentally demonstrate the generation and characterization of a micron-scale optical hexagonal lattice with well defined orbital and radial numbers in each unit cell. The described methods provide a robust and general method of generating and controlling structured waves such as optical skyrmions and matter wave implementations of orbit and spin-orbit beams.

Figures (3)

• Figure 1: Example of k-space coherent averaging to generate a hexagonal lattice with $\ell=1$ & $n_r=2$. a) The first step of the lattice-generation method, where an initial on-axis input beam at (0,0) is transformed through successive applications of the branching operator $\mathcal{B}$, producing six beams in the first iteration and 19 unique beams (out of 36 total) after the 2nd iteration of the $\mathcal{B}$. Circle size indicates the relative amplitude. b) Polarization encoding of the 19 branched beams, where each beam is assigned a polarization direction corresponding to a spin–orbit state of $\ell=1$ and $n_r=2$. (c) Resulting real-space intensity profile, post-selected on left-circular polarization (LCP), obtained from the coherent superposition of the polarization-encoded beams in (b). (d) Corresponding phase distribution, and the inset shows a single lattice site from the phase map. The azimuthal phase winding of $2\pi$ illustrates $\ell=1$, while the presence of two radial phase reversals indicates the $n_r=2$. e) State of polarization distribution for a unit cell, showing how polarization varies spatially along the lattice site.
• Figure 2: Schematic of the proposed setup for generating neutron spin–orbit lattices with orbital angular momentum $\ell=1$ and $n_r=1$ using a five-blade neutron interferometer (NI). An incident neutron beam is coherently split by the first beam splitter (BS1) into two paths, which are further divided by the second beam splitter (BS2) to form four distinct beams. Each of the four paths in Configuration 1 (for the square lattice) contains a magnetic prism oriented at $0^\circ$, $45^\circ$, $90^\circ$, and $135^\circ$, that imparts controlled angular deflection and spin rotation to encode the spin–orbit coupling. In Configuration 2, only three prisms are used, oriented at $0^\circ$, $60^\circ$ and $120^\circ$, while the fourth path is blocked to produce the hexagonal lattice geometry. The beams are redirected and recombined by subsequent beam splitters (BS3–BS5), with the neutron's wave function forms a superposition of the four paths that constitutes a lattice of spin–orbit states. After the final recombination at BS5, the output beam is analyzed by a spin analyzer, and the resulting two-dimensional intensity distribution is recorded on a neutron detector.
• Figure 3: (a) Schematic of the experimental setup, implementing the first iteration of the beam-branching operator, for independent control over the phase, polarization, and wave-vector configuration of the interfering beams. A 642 nm circularly polarized laser beam is successively split and recombined through successive stages of an interferometer(BS1–BS12, M1–M6), generating a coherent superposition of six beams. Linear polarizers in each beam path provide independent control of the polarization of each interfering beam, while mirrors mounted on kinematic stages allow adjustment of the wave-vector configuration, relative phase, and spatial overlap. To analyze polarization-dependent interference effects, the output beams are passed through a quarter-wave plate and a linear polarizer before being recorded by a CMOS camera at the imaging plane. (b) Simulated and observed spin–orbit lattices. Column 1: top row shows the simulated intensity profile of six beams following the first iteration of the branching operator, post-selected on left-circular polarization, while the bottom row shows the corresponding overlay of the phase map and polarization ellipse distribution for a unit cell. Columns 2 and 3: top row shows experimentally observed intensity profiles for lattice periods of 158 $\mu$m and 50 $\mu$m, respectively, while the bottom row shows the corresponding phase variations, illustrating an OAM of $\ell = 1$, overlaid with polarization ellipse distributions extracted from the experimental data using Stokes parameter analysis.