🔬 optics

Optimal Diffractive Focusing of Vortex Beams

This paper formulates and solves an exact analytical variational problem to determine the optimal radial pupil for maximizing the ring intensity of vortex beams through a finite aperture, establishing fundamental performance limits and experimentally validating the theory against practical diffractive and refractive focusing methods.

Original authors: Roohollah Ghobadi, Carrie Rodriguez, Lukas Scarfe, Maxim A. Efremov, Ebrahim Karimi

Published 2026-10-09

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Roohollah Ghobadi, Carrie Rodriguez, Lukas Scarfe, Maxim A. Efremov, Ebrahim Karimi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Light, sound, and even the waves of matter that make up the physical world share a common challenge: how to gather their energy into a tight, powerful spot. For centuries, scientists have solved this by using glass lenses or curved mirrors to bend waves so they arrive at a single point in perfect unison. This works beautifully for ordinary light, but it hits a wall when the waves carry a special kind of twist called orbital angular momentum. These "vortex" beams are shaped like tiny tornadoes, with a hollow center where the intensity drops to zero. Because of this hollow core, they cannot be focused into a single bright point like a laser pointer. Instead, they naturally want to concentrate their energy into a ring, much like the eye of a hurricane. The question researchers have long faced is how to shape these twisting beams to make that ring as bright and tight as possible, especially when traditional glass lenses are too difficult or impossible to use, such as in the realm of X-rays or electron beams.

A team of physicists has now mapped out the absolute best way to do this, establishing a new rulebook for focusing these swirling beams. They approached the problem not by guessing which lens shape might work, but by treating it as a mathematical puzzle to find the single most efficient pattern of light. Imagine trying to squeeze the most water out of a sponge; there is a specific way to press it that yields the maximum amount. Similarly, these researchers calculated the exact pattern of light amplitude—the brightness across the beam's face—that would concentrate the most energy onto a specific ring at a specific distance. They discovered that the ideal pattern is not just a simple set of rings, but a complex combination of two distinct features. One part handles the timing of the waves to ensure they arrive together, while the other part shapes the brightness in a way that depends entirely on how many twists the beam carries.

The researchers found that for a beam with no twist, the best focusing pattern is a familiar one, similar to the concentric rings found in a classic Fresnel zone plate, a type of flat lens made of alternating clear and dark rings. However, as soon as the beam carries a twist, the optimal pattern changes fundamentally. The brightness must now follow a specific curve that oscillates in a way that matches the beam's spin, described mathematically by a Bessel function. If one were to ignore this requirement and simply use a standard ring pattern, the resulting focus would be significantly dimmer. The team proved that the theoretical limit for how bright this ring can get is higher than what standard designs achieve, and they identified the precise mathematical structure needed to reach that limit. This structure combines a wave-like oscillation with a radial modulation that features multiple lobes, which get wider as the twist of the beam increases.

To verify that their calculations matched reality, the team built an experiment using a computer-controlled screen that could manipulate light. They programmed this screen to create the exact patterns their theory predicted, including both the ideal, continuous pattern and simplified, "binary" versions that only used two levels of brightness, similar to a black-and-white image. They then sent laser beams carrying different amounts of twist through these patterns and watched how they traveled through the air. The results confirmed their theory with striking precision. The beams shaped by the ideal pattern formed the brightest possible rings, while the simplified binary versions, though still effective, fell short of the maximum potential. The experiment showed that the difference between a good focus and the best possible focus lies in how well the brightness is distributed across the beam to match its internal twist.

This work does more than just improve how we focus a specific type of laser; it sets a fundamental benchmark for all future devices that need to handle these twisted waves. In fields ranging from high-resolution microscopy to the manipulation of atoms and electrons, where traditional glass lenses cannot be used, knowing the theoretical limit of performance is crucial. The researchers demonstrated that by understanding the interplay between the wave's twist and its shape, scientists can design optical elements that come much closer to the physical limits of what is possible. Their findings provide a clear guide for engineers and physicists: to get the most out of a vortex beam, one must not only manage the timing of the waves but also carefully sculpt the brightness of the beam to match its unique rotational character. This ensures that when these powerful, swirling beams are finally brought to a focus, they deliver the maximum possible intensity for the task at hand.

Technical Summary: Optimal Diffractive Focusing of Vortex Beams

Problem Statement
Conventional focusing aims to concentrate wave energy into a single point using refractive elements (lenses) or mirrors. However, for waves carrying orbital angular momentum (OAM), known as vortex beams, the helical phase structure ( $e^{i\ell\phi}$ ) creates a phase singularity on the optical axis, forcing the intensity to vanish at the center for any non-zero topological charge $\ell$ . Consequently, the natural focusing objective for these beams is not a point, but the concentration of energy onto an annular ring. While diffractive optical elements (such as Fresnel zone plates) offer lens-free alternatives for focusing, the fundamental limits of diffractive focusing for OAM-carrying beams have not been rigorously established. Specifically, it is unknown what radial pupil function maximizes the intensity at a prescribed ring radius and propagation distance under normalization constraints, nor how this optimal performance compares to standard binary or amplitude-weighted designs.

Methodology
The authors formulate the diffractive ring focusing of paraxial vortex beams as a variational optimization problem.

Mathematical Framework: The input field is defined as $\Psi_0(\rho, \phi) = \psi_\ell(\rho)e^{i\ell\phi}$ within a finite circular aperture. The propagation is modeled using the paraxial wave equation (equivalent to a free-particle Schrödinger equation), leading to an integral representation of the field involving Bessel functions.
Optimization Goal: The objective is to maximize the intensity functional $J[\psi_\ell] = |\Psi(\rho^*, \tau_f)|^2$ at a target ring radius $\rho^*$ and propagation distance $\tau_f$ , subject to a normalization constraint on the input field energy.
Variational Approach: Using a Lagrange multiplier, the problem is transformed into an integral-kernel eigenvalue problem. The kernel $K_\ell(\rho, \rho')$ encodes the propagation dynamics and the target geometry.
Analytical Solution: The problem admits an exact analytical solution where the optimal pupil function corresponds to the principal eigenfunction of the kernel. The maximum achievable intensity is determined by the largest eigenvalue.
Comparative Analysis: The theoretical framework is used to derive efficiency metrics for practical mask designs, specifically binary Fresnel–Bessel zone plates and amplitude-weighted Fresnel–Bessel masks, comparing their performance against the theoretical upper bound.
Experimental Validation: The theory is tested in the optical domain using a spatial light modulator (SLM) to encode the calculated pupil functions. A linearly polarized Gaussian beam is modulated, filtered, and propagated, with intensity distributions recorded at various axial positions to verify the ring formation and peak intensity.

Key Contributions and Results

Exact Analytical Solution: The paper derives the exact form of the optimal radial pupil function. The solution reveals that the optimal pupil combines a familiar Fresnel phase structure with an OAM-dependent radial modulation described by a Bessel function ( $J_\ell$ ). The associated eigenvalue provides a rigorous upper bound on the achievable ring intensity.
Fundamental Limits: The study establishes that for $\ell \neq 0$ , optimal focusing requires two distinct components: Fresnel phase matching (to focus at the correct distance) and specific radial apodization (to match the vortex geometry).
Performance of Practical Masks:
- Binary Fresnel–Bessel Masks: When both the Fresnel phase and the Bessel amplitude are binarized (sign-only), the efficiency suffers a penalty. In the small-argument regime, the efficiency is reduced by a factor of $4(\ell+1)/(\ell+2)^2$ compared to the optimal solution. In the large-argument regime, further reduction occurs due to the binarization of the Bessel oscillations.
- Amplitude-Weighted Masks: If the Bessel radial apodization is preserved while only the Fresnel oscillation is binarized, the efficiency recovers the universal binary-harmonic limit ( $8/\pi^2$ ) known for $\ell=0$ beams. This indicates that the OAM-specific penalty arises solely from the improper radial weighting, not the phase quantization.
Experimental Confirmation: Experiments confirm that amplitude-weighted Fresnel–Bessel masks yield systematically higher ring intensities than fully binary masks. The observed vortex-ring evolution closely matches theoretical predictions, validating the derived pupil structures.

Significance
The paper establishes the vortex-beam analogue of optimal amplitude-only focusing, providing a rigorous benchmark for evaluating diffractive vortex optics. It clarifies that the interplay between Fresnel diffraction and the radial structure imposed by OAM dictates the ultimate limits of intensity concentration. By identifying the specific role of Bessel-type radial modulation, the work offers a systematic framework for designing high-efficiency ring-focusing elements. This is particularly relevant for applications requiring structured wave concentration, such as stimulated emission depletion microscopy, nonlinear optics, and strong-field physics. Furthermore, because the formulation relies only on aperture-limited propagation, the results extend beyond optics to other wave systems, including electron and neutron beams, where conventional refractive optics are difficult to realize.

More like this