🔬 optics

Passive Cross-Basis Mode Transitions Along a Single Freely Propagating Bessel Beam

This paper demonstrates that by encoding discrete transverse modes into independent annular regions of a single static spatial light modulator, a freely propagating Bessel beam can passively transition through various complex optical modes (such as vortex, Hermite-Gaussian, and Airy) at specific axial distances without requiring dynamic modulation or cascaded optical elements.

Original authors: Henry P. Evans, Layton A. Hall

Published 2026-05-19

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CC BY 4.0

Original authors: Henry P. Evans, Layton A. Hall

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

Imagine you have a flashlight that usually shines a single, unchanging pattern of light. If you want to change that pattern, you typically have to stop, swap out a lens, or use a complex machine to reshape the beam.

This paper describes a clever trick that lets a single beam of light change its shape automatically as it travels forward, without any moving parts or extra lenses.

Here is how it works, using simple analogies:

The "Magic Slide" Analogy

Think of a standard laser beam like a train moving down a track. Usually, the train cars (the light pattern) are fixed. If you want the train to look different at the next station, you have to rebuild the train while it's stopped.

The researchers found a way to build a "magic slide" (a special screen called a Spatial Light Modulator, or SLM) that the light passes through just once. This slide is divided into concentric rings, like a target or a tree stump with rings.

The Rule: The researchers discovered a strict rule for Bessel beams (a special type of light): Where the light starts on the ring determines where it lands on the track.
- Light starting on the inner rings of the slide lands closer to the source.
- Light starting on the outer rings of the slide lands further away.

The "Concert Hall" Analogy

Imagine a concert hall where the stage is the light source, and the audience is the space the light travels through.

Normally, if you want the music to sound like a Jazz band at the front row and a Rock band at the back row, you'd need two different bands playing at different times or different speakers.
In this experiment, the researchers put one giant speaker (the SLM) at the front.
- The inner part of the speaker plays a "Jazz" pattern (a Bessel beam).
- The middle part plays a "Rock" pattern (a Vortex beam).
- The outer part plays a "Classical" pattern (an Airy beam).

Because of the special physics of these light beams, the "Jazz" sound only becomes clear to the audience sitting close to the stage. The "Rock" sound only becomes clear to the audience sitting in the middle. The "Classical" sound only becomes clear to the people at the very back.

As the light travels down the "hall," it passively changes its identity. You don't need to switch speakers or move the band; the light simply reveals different patterns at different distances.

What They Actually Did

The team used a computer-controlled screen (the SLM) to paint different patterns on these rings:

Zone 1 (Close): Created a standard, solid ring of light (Bessel mode).
Zone 2 (Middle): Created a ring of light that spins like a tornado (Bessel Vortex).
Zone 3 (Further): Created a pattern that looks like a grid of squares (Hermite-Gaussian).
Zone 4 (Furthest): Created a pattern that curves and accelerates like a sliding wave (Airy).

They fired a single beam of red light through this screen. As the beam traveled 25 millimeters through the air, they took pictures at different distances.

At 3mm, they saw the first pattern.
At 8mm, the first pattern faded, and the second pattern appeared.
At 14mm, the third pattern appeared.
At 20mm, the fourth pattern appeared.

Why It's Special

Usually, changing from a "square" light pattern to a "spinning" light pattern requires complex, active machines or stacking many lenses. This paper shows that you can do it with one static screen and no moving parts. The light changes its shape simply by traveling forward, thanks to the geometry of how Bessel beams are built.

The Results

They proved this works with high accuracy. When they checked the light at the specific distances, the patterns matched their computer designs about 84% to 96% of the time, and there was very little "crosstalk" (meaning the "Jazz" didn't leak into the "Rock" section).

In short: They turned a single beam of light into a "programmable movie" where the scene changes automatically as the light moves forward, all controlled by a single, static piece of glass.

Technical Summary: Passive Cross-Basis Mode Transitions Along a Single Freely Propagating Bessel Beam

Problem Statement
Transverse spatial modes of optical beams (e.g., Hermite–Gaussian, Laguerre–Gaussian, Airy, and Bessel families) represent distinct orthogonal state spaces defined by unique underlying spatial symmetries: Cartesian, cylindrical, caustic, and propagation-invariant, respectively. Traditionally, these modal families are treated as separate bases that do not interconvert through simple free-space propagation. While transitions between modal families are possible, they typically require dedicated active or passive optical elements, such as astigmatic mode converters, multi-plane light conversion (MPLC), cascaded diffractive elements, metasurfaces, or dynamically reconfigurable spatial light modulators (SLMs). A significant limitation in existing literature is that modal evolution usually remains within a single transverse symmetry class (e.g., "frozen waves" or $\ell$ -stepping) or, when crossing classes, necessitates complex setups involving polarization-sensitive subwavelength elements, cascaded optics, or dynamic modulation. Consequently, there is no established method for achieving passive, deterministic transitions between incompatible transverse symmetry classes along a single beam without dynamic modulation or cascaded elements.

Methodology
The authors propose a method that leverages the conical angular spectrum of Bessel beams to establish a one-to-one mapping between radial beam position and axial reconstruction distance. This geometric correspondence, inherent to axicon-generated Bessel fields, is utilized to transform the radial aperture of a single static, phase-only, polarization-insensitive SLM into a programmable longitudinal-mode register.

The core mechanism involves partitioning the SLM aperture into independent concentric annular regions. Each annulus is assigned a distinct phase modulation function ( $f_m$ ) corresponding to a specific transverse mode. The total phase profile $\Phi(r_s, \phi)$ on the SLM is defined as:
$\Phi(r_s, \phi) = k_r r_s + \sum_m f_m(r_s, \phi) \Pi\left(\frac{r_s - r_{m,s}}{\Delta r}\right)$
where $k_r$ is the radial spatial frequency fixing the Bessel cone angle, $r_{m,s}$ is the central radius of the $m$ -th annulus, and $\Delta r$ is its width.

The axial reconstruction position $z_m$ for a specific annular zone is determined by the paraxial relation $z_m \simeq r_{m,s} k / k_r$ . This allows the encoding of different transverse mode families at preselected axial planes. The axial extent ( $\Delta z$ ) of each mode is controlled by the radial thickness of the annulus ( $\Delta r$ ). Because the mapping is purely geometrical, any transverse mode expressible as a phase pattern (including HG, LG, Airy, and Bessel modes) can be assigned to an independent axial register within a single static hologram.

Key Contributions

Passive Longitudinal Modal Register: The work demonstrates that the longitudinal propagation coordinate can function as an addressable modal degree of freedom, encoded entirely within a single static, phase-only SLM.
Cross-Basis Transitions: The paper achieves passive transitions between fundamentally distinct transverse symmetry classes (Cartesian, cylindrical, caustic, and propagation-invariant) without dynamic modulation or cascaded optical elements.
Programmable Topology: The method allows for the independent programming of the number of lattice sites and the azimuthal topology of the intensity landscape at different axial depths.

Experimental Results
The authors validated the principle using a 635 nm continuous-wave laser directed onto a phase-only SLM, followed by a 4f imaging system with 4× demagnification onto a translating CCD camera.

Ring-Lattice Fields: The SLM was partitioned to encode ring-lattice patterns with varying site numbers ( $N$ ). Measurements confirmed that the number of lattice sites could be independently programmed at specific axial depths (e.g., $N=1$ at 3 mm, $N=6$ at 8 mm, $N=8$ at 14 mm, and $N=12$ at 20 mm).
Cross-Basis Modal Transitions: A single beam was encoded to sequentially transition through four distinct modal families at specific axial planes:
- $z = 5$ mm: Zero-order Bessel-Gaussian (Propagation-invariant).
- $z = 10$ mm: Bessel vortex with topological charge $\ell=4$ (Cylindrical).
- $z = 15$ mm: Hermite-Gaussian-Bessel (HGB) (Cartesian).
- $z = 22$ mm: Airy caustic mode (Caustic).
Performance Metrics: The experimental results showed high fidelity with theoretical calculations. The intensity fidelity (overlap integral) was measured at 96%, 84%, 94%, and 91% for the four respective planes. Crosstalk between regions was measured to be less than 5%.

Significance and Claims
The paper claims that the conical angular spectrum of Bessel beams provides a natural physical mechanism for cross-modal changes during propagation. By utilizing the geometric mapping between radial launch radius and axial reconstruction distance, the authors establish a passive method for generating beams with evolving modal and topological states.

The significance of this work lies in its ability to create volumetric optical structures where the modal identity changes discretely with the propagation distance $z$ . The authors identify direct implications for:

Volumetric Optical Manipulation: Enabling the simultaneous addressing of different particle species or trapping geometries at distinct axial depths without mechanical refocusing.
Axially Multiplexed Communications: Offering a complementary degree of freedom to transverse orbital angular momentum (OAM) multiplexing, potentially supporting range-selective receivers.
Three-Dimensional Photonic State Preparation: Providing a route to prepare 3D photonic states without cascaded elements, with an encoding principle that is wavelength-scalable and compatible with ultrafast pulse fronts.

The authors emphasize that this approach does not rely on dynamic modulation or complex cascaded optics, representing a shift from active or multi-element solutions to a single static element capable of complex longitudinal modal engineering.