OrthoVortex: Fast Vortex Beam Alignment

• OrthoVortex is a vortex beam alignment framework that restores modal orthogonality in LOS MIMO networks using cross-modal phase signatures.
• It employs a few-shot estimation technique to correct angular misalignment in OAM mode multiplexing, reducing reliance on exhaustive scans.
• Experimental validation at 120 GHz showed a 12.08 dB SIR gain and a 4.57× capacity improvement, demonstrating its practical impact on high-capacity LOS links.

OrthoVortex is a fast vortex beam alignment framework for orbital angular momentum (OAM) mode multiplexing in line-of-sight (LOS) MIMO networks. It is designed for the regime in which OAM-based communication systems offer high-capacity multiplexing in LOS scenarios, but their performance is sensitive to nodal misalignment, which disrupts modal orthogonality and hinders the data multiplexing gain. The framework estimates misalignment angles and applies an appropriate phase correction to restore orthogonality between modes. In contrast to prior approaches based on impractical fully digital arrays or exhaustive beam scans, OrthoVortex introduces the cross-modal phase as a unique signature for identifying misalignment angles and implements a few-shot alignment technique intended to be feasible for real-world RF systems (Mollahosseini et al., 26 Aug 2025).

1. Conceptual basis and communication setting

OrthoVortex is situated in LOS channels at mmWave and sub-THz frequencies, where propagation is sparse and usually dominated by a single path. In such settings, conventional spatial multiplexing gains are limited, whereas OAM beams provide a way to multiplex multiple independent data streams under LOS by exploiting modal orthogonality (Mollahosseini et al., 26 Aug 2025).

An OAM beam has a helical phase front whose azimuthal phase winds an integer number of times around the beam axis, creating a central phase singularity and a donut-shaped intensity profile. The OAM mode is indexed by an integer $l$, the topological charge, with azimuthal phase

$\exp(j l \phi),$

where $\phi$ is the azimuthal angle around the beam axis. Modal orthogonality follows from

$$\int_{0}^{2\pi} e^{j l \phi}\, e^{-j l' \phi}\, d\phi \;=\; 2\pi\, \delta_{l,l'},$$

which ensures no mutual interference when distinct $l$ are transmitted simultaneously (Mollahosseini et al., 26 Aug 2025).

The framework models OAM generation and reception with uniform circular arrays (UCAs) and metasurface-based vortex beams at RF. For UCAs, the $n$th transmit element at azimuth $\phi_n = \frac{2\pi n}{N_t}$ applies a progressive phase $\exp(j l \phi_n)$ to generate mode $l$. The receive side is also a UCA, or emulated apertures, and the analysis uses far-field approximations together with a large-$N_t$ summation-to-integral approximation that yields Bessel-function-based closed forms typical in circular-array OAM formulations (Mollahosseini et al., 26 Aug 2025).

2. Misalignment model and loss of modal orthogonality

OrthoVortex focuses on receive-side angular misalignment. The misalignment parameters are the elevation $\exp(j l \phi),$0 and azimuth $\exp(j l \phi),$1 of the transmit center as seen in the receive frame. Translational misalignment can be beam-steered at the transmitter in many LOS use cases, whereas angular misalignment at the receiver causes a spatial phase gradient across the aperture that distorts the helical phase, breaks orthogonality, and induces inter-modal interference (IMI) (Mollahosseini et al., 26 Aug 2025).

The received discrete-mode signal in a multiplexed OAM system is modeled as

$\exp(j l \phi),$2

where $\exp(j l \phi),$3 stacks the symbols per OAM mode, $\exp(j l \phi),$4 stacks the mode-matched receive outputs, $\exp(j l \phi),$5 is noise, and $\exp(j l \phi),$6 is the mode-coupling matrix. Under perfect alignment, $\exp(j l \phi),$7 is approximately diagonal due to modal orthogonality. Misalignment introduces off-diagonal terms $\exp(j l \phi),$8, reflecting leakage between modes (Mollahosseini et al., 26 Aug 2025).

For transmit mode $\exp(j l \phi),$9 and wavenumber $\phi$0, the per-antenna complex baseband received signal at the $\phi$1th receive antenna is derived as

$\phi$2

where $\phi$3 collects constants, $\phi$4 is the Tx–Rx center separation vector, and $\phi$5 are array-element positions. After far-field amplitude and phase approximations, first-order Taylor expansion of path length, and a large-$\phi$6 summation-to-integral approximation, this becomes

$\phi$7

with

$\phi$8

Equation (2) separates a mode-independent term, which functions as a global plane-wave tilt factor, from a mode-dependent term carrying $\phi$9. This decomposition is the physical basis of OrthoVortex. A plausible implication is that the alignment problem can be reduced from generic beam search to extraction of a structured, geometry-dependent phase signature (Mollahosseini et al., 26 Aug 2025).

3. Cross-modal phase and angle identifiability

The central construct introduced by OrthoVortex is the cross-modal phase, defined as the relative phase between received signals for two OAM modes at the same receive antenna. For a single tone $$\int_{0}^{2\pi} e^{j l \phi}\, e^{-j l' \phi}\, d\phi \;=\; 2\pi\, \delta_{l,l'},$$0,

$$\int_{0}^{2\pi} e^{j l \phi}\, e^{-j l' \phi}\, d\phi \;=\; 2\pi\, \delta_{l,l'},$$1

The squaring operation cancels the possible $$\int_{0}^{2\pi} e^{j l \phi}\, e^{-j l' \phi}\, d\phi \;=\; 2\pi\, \delta_{l,l'},$$2 phase flips from the real Bessel term. As a result, mode-independent factors, including the unknown distance $$\int_{0}^{2\pi} e^{j l \phi}\, e^{-j l' \phi}\, d\phi \;=\; 2\pi\, \delta_{l,l'},$$3 and absolute phase offsets, cancel, leaving a signature determined only by $$\int_{0}^{2\pi} e^{j l \phi}\, e^{-j l' \phi}\, d\phi \;=\; 2\pi\, \delta_{l,l'},$$4 and $$\int_{0}^{2\pi} e^{j l \phi}\, e^{-j l' \phi}\, d\phi \;=\; 2\pi\, \delta_{l,l'},$$5 (Mollahosseini et al., 26 Aug 2025).

In LOS conditions, the cross-modal phase is frequency-invariant because the same tilt-induced geometric phase affects both modes equally. This permits multi-tone averaging across subcarriers: $$\int_{0}^{2\pi} e^{j l \phi}\, e^{-j l' \phi}\, d\phi \;=\; 2\pi\, \delta_{l,l'},$$6 with $$\int_{0}^{2\pi} e^{j l \phi}\, e^{-j l' \phi}\, d\phi \;=\; 2\pi\, \delta_{l,l'},$$7 including AWGN (Mollahosseini et al., 26 Aug 2025).

For a given antenna $$\int_{0}^{2\pi} e^{j l \phi}\, e^{-j l' \phi}\, d\phi \;=\; 2\pi\, \delta_{l,l'},$$8, $$\int_{0}^{2\pi} e^{j l \phi}\, e^{-j l' \phi}\, d\phi \;=\; 2\pi\, \delta_{l,l'},$$9 is analytically known from equation (3). Measuring $l$0 for at least three antennas that are not diametrically opposite and across two modes creates a system that determines $l$1, $l$2, and $l$3. To remove the $l$4-ambiguity in $l$5, OrthoVortex tests phase masks computed with $l$6 and $l$7 and selects the one yielding higher post-correction received power (Mollahosseini et al., 26 Aug 2025).

The measurement protocol is deliberately sparse. Minimal measurements require two transmitted modes, such as $l$8 and $l$9, and three receive antennas, corresponding to six shots with one RF chain. With $n$0 parallel RF chains, acquisition can proceed concurrently, reducing time linearly. Because the method uses phase-only differences, amplitude calibration is not required (Mollahosseini et al., 26 Aug 2025).

4. Estimation, correction, and few-shot operation

With redundancy over $n$1 modes, $n$2 antennas, and $n$3 subcarriers, OrthoVortex estimates the angles through the optimization

$n$4

where $n$5 weights antennas by received amplitude to favor higher-SNR elements, $n$6 is the selected antenna set, and $n$7 is the selected mode set (Mollahosseini et al., 26 Aug 2025).

Once $n$8 are obtained, the receiver applies a per-element phase mask that cancels the tilt-induced phase gradient. For the receive element at Cartesian coordinates $n$9,

$\phi_n = \frac{2\pi n}{N_t}$0

This linear phase ramp corresponds to a plane wave arriving from $\phi_n = \frac{2\pi n}{N_t}$1. When superimposed with the mode-matching conjugate phase for the desired mode, it flattens the distorted helical wavefront so that modal orthogonality is restored and IMI is minimized, making $\phi_n = \frac{2\pi n}{N_t}$2 more diagonal (Mollahosseini et al., 26 Aug 2025).

The operational sequence is a few-shot protocol: pilot transmission on selected OAM modes and subcarriers, data collection on selected antennas, cross-modal phase estimation, solution of the low-dimensional optimization problem, and programming of the receive-side phase shifters. The problem dimension is restricted to $\phi_n = \frac{2\pi n}{N_t}$3, and evaluating the loss scales as $\phi_n = \frac{2\pi n}{N_t}$4 per iteration. This suggests that computational burden is secondary to measurement fidelity in the intended deployment regime (Mollahosseini et al., 26 Aug 2025).

The framework is explicitly positioned against two alternatives. Fully digital arrays combined with MUSIC or ESPRIT require element-level digitization at D-band and are described as impractical due to cost and power. Exhaustive or hierarchical beam scans incur latency proportional to grid resolution and are unsuitable for fast dynamics. OrthoVortex instead uses mode physics through the cross-modal phase and is compatible with hybrid MIMO receivers (Mollahosseini et al., 26 Aug 2025).

5. Hardware realization, bandwidth, and design guidelines

The experimental realization uses over-the-air measurements at 120 GHz with low-cost, rapidly prototyped metasurfaces. The RF chain comprises IF generation with a Keysight M8195A, local oscillator generation with a Keysight E8257D, and a $\phi_n = \frac{2\pi n}{N_t}$5 up-converter, VDI WR6.5CCU-M4, producing approximately 116 GHz LO and mixed to 120 GHz. The transmitter horn illuminates a passive transmissive metasurface engineered to impose $\phi_n = \frac{2\pi n}{N_t}$6 per element using C-shaped split-ring resonators in hot-stamped aluminum on paper. The metasurface forms five concentric rings with radii approximately $\phi_n = \frac{2\pi n}{N_t}$7–$\phi_n = \frac{2\pi n}{N_t}$8 cm and $\phi_n = \frac{2\pi n}{N_t}$9–$\exp(j l \phi_n)$0 elements per ring. The receiver is a virtual UCA emulated by translating a single probe to 20 positions at radius $\exp(j l \phi_n)$1 mm, and misalignment is created by rotating the virtual array in three dimensions (Mollahosseini et al., 26 Aug 2025).

The demonstrations use a short-range 40 cm receive-plane scan, which is near-field relative to the Fraunhofer limit $\exp(j l \phi_n)$2 m. The framework is stated to remain valid because near-field deviations induce common-mode phase that cancels in the cross-modal phase. Multi-tone processing spans 119.5–120.2 GHz with 71 subcarriers at 10 MHz spacing, and the metasurface response is flat across this bandwidth (Mollahosseini et al., 26 Aug 2025).

Two practical dependencies receive explicit treatment. Increasing the number of subcarriers $\exp(j l \phi_n)$3 improves estimation by averaging frequency-invariant cross-modal phase observations, reducing variance without time overhead. Increasing antenna count $\exp(j l \phi_n)$4 adds spatial redundancy; the reported behavior is that accuracy improves markedly up to $\exp(j l \phi_n)$5 and then saturates, suggesting $\exp(j l \phi_n)$6 as a sweet spot between accuracy and time cost (Mollahosseini et al., 26 Aug 2025).

The paper also states several design rules. Mode selection should favor the two modes with the highest expected received power, accounting for divergence and aperture-distance geometry. Antenna selection should maximize azimuthal separation while avoiding diametrically opposite pairs. Angle estimation should restrict $\exp(j l \phi_n)$7, and $\exp(j l \phi_n)$8 modulo $\exp(j l \phi_n)$9 should be resolved through mask-performance selection. These rules indicate that OrthoVortex is not a generic black-box estimator but a geometry-aware alignment method (Mollahosseini et al., 26 Aug 2025).

6. Experimental performance, operating assumptions, and limitations

The principal reported results combine simulation and over-the-air measurement. Using $l$0, $l$1, a single tone at 120 GHz, distances of approximately 40 cm, and misalignment up to $l$2 with $l$3, OrthoVortex achieves mean absolute error of approximately $l$4 for azimuth and $l$5 for elevation (Mollahosseini et al., 26 Aug 2025). The paper notes that smaller $l$6 yields larger error because the sensitivity of $l$7 to $l$8 decreases near $l$9.

Performance after correction is quantified by the signal-to-interference ratio

$N_t$0

and by the low-noise, IMI-limited capacity

$N_t$1

Applying the receive phase mask computed from the estimated angles yields an average SIR gain of $N_t$2 dB across the tested misalignments and an average $N_t$3 improvement in capacity. IMI maps show that post-correction energy concentrates back on the transmitted mode, suppressing off-diagonal leakage (Mollahosseini et al., 26 Aug 2025).

The implementation is described as the first-ever experimental validation of OAM beam alignment with RF transceivers. With three receive RF chains, the protocol becomes a four-shot procedure, using two transmit slots and two receive slots, which the paper characterizes as suitable for fast alignment in mobile LOS links (Mollahosseini et al., 26 Aug 2025).

The method also carries explicit assumptions and limitations. It assumes LOS dominance and sparse multipath, conditions under which cross-modal phase remains frequency-invariant and insensitive to distance. The derivation uses far-field approximations, although the experiments indicate robustness in near-field meter-scale settings because common-mode phase cancels in the cross-modal phase. The current framework addresses receive-side angular misalignment only; general two-sided tilt would require cooperative estimation of four angles. Severe mispointing that misses the receive aperture prevents estimation because SNR becomes insufficient. Wideband effects such as beam squint are minimal over less than 1 GHz around 120 GHz, but larger bandwidths may require per-subcarrier masks (Mollahosseini et al., 26 Aug 2025).

A common misconception is that OAM alignment must rely on exhaustive beam search or fully digital element-level observation. OrthoVortex directly contradicts that assumption by showing that a small number of mode-indexed pilot measurements, interpreted through the cross-modal phase, is sufficient to estimate misalignment and restore modal orthogonality in the reported LOS sub-THz setting (Mollahosseini et al., 26 Aug 2025).