🔬 applied physics

Single Plane Spatial Mode Sorter

This paper presents and experimentally validates an optimal single-plane device capable of sorting diverse orthogonal spatial modes (including Hermite-Gaussian, Laguerre-Gaussian, and Bessel-Gaussian) into separate output channels with minimal cross-talk, while also enabling arbitrary mode generation in reverse and achieving the theoretical 1/M power transmission limit.

Original authors: Khen Cohen, Yoav Yosif-Or, Yaron Oz, Ady Arie

Published 2026-04-16

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Khen Cohen, Yoav Yosif-Or, Yaron Oz, Ady Arie

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 are at a crowded party where everyone is wearing a different colored shirt, but they are all standing in a jumbled pile. Your goal is to separate them so that everyone in a red shirt ends up in one corner, everyone in blue in another, and so on, without anyone getting mixed up.

In the world of light, "shirts" are called spatial modes. These are specific patterns that light waves can take (like ripples on a pond). Scientists use these patterns to carry huge amounts of information, much like how different radio stations carry different songs. But to read the message, you need a machine that can instantly sort these light patterns into their own separate "corners."

This paper introduces a new, incredibly smart machine to do exactly that. Here is the breakdown in simple terms:

1. The Problem: The "Sorting Hat" is Too Heavy

Previously, sorting these light patterns was like trying to organize that party crowd using a complex series of mirrors and lenses. You needed a stack of 4 to 10 layers of glass (like a multi-story building) to get the job done right.

The Issue: These machines were bulky, hard to build, and if you wanted to sort many different patterns, the machine got huge and complicated.
The Old "Fork" Trick: There was an old trick called a "Fork Grating" that worked well for one specific type of light pattern (twisting light), but it failed miserably if the patterns were slightly different.

2. The Solution: A Single "Magic Sheet"

The authors created a Single Plane Spatial Mode Sorter.

The Analogy: Imagine replacing that multi-story building with a single, transparent sheet of plastic (like a high-tech sticker) that you stick on a window.
How it works: When the mixed-up light hits this sheet, the sheet acts like a magical traffic director. It instantly redirects every specific light pattern to its own unique exit door.
The Result: It works for many different types of light patterns (not just the twisting ones), and it does it with almost zero mistakes (cross-talk).

3. The Catch: The "Pizza Slice" Trade-off

There is one price to pay for using this single sheet instead of a multi-layer building.

The Analogy: Imagine you have a pizza (the total light energy). If you have to feed 4 people (4 different light modes), and you use this new sorter, you have to cut the pizza into 4 slices. Each person only gets 1/4 of the pizza. If you have 100 people, they each get a tiny crumb.
The Science: The more patterns you try to sort at once, the dimmer the light gets at the exit. The paper proves mathematically that this is the best possible outcome for a single-layer device. You can't cheat physics; you can't get a full pizza out of a single slice. However, for many applications (like secure quantum communication), having a tiny bit of light is better than having no light at all.

4. Two-Way Street: Sorting and Making

This device is unique because it works in reverse.

Sorting: You throw mixed light in, and it sorts them out.
Generating: If you shine a simple, plain beam of light backwards through the sorter, it magically transforms into a complex, specific pattern. It's like a reverse printer that can print any shape of light you want just by feeding it a blank sheet.

5. Real-World Superpowers

The paper shows this isn't just theory; they built it and tested it.

Robustness: It's tough. If the light source wobbles a little or the temperature changes slightly, the sorter still works. It's like a good pair of glasses that still lets you see even if you tilt your head.
Spectroscopy (The "Rainbow Ruler"): Because the sorter is so sensitive to the color (wavelength) of the light, the authors realized they could use it as a super-precise ruler. If the light color changes by a tiny amount, the light hits a slightly different spot on the detector. This allows them to measure the exact "color" of a chemical or gas just by seeing where the light lands.

Summary

Think of this new device as a universal, single-layer traffic cop for light.

Before: You needed a massive, complex highway interchange to sort different cars.
Now: You have a single, smart traffic light that directs every car to its lane instantly.
The Trade: The cars have to slow down a bit (less light intensity), but they get to their destination much faster and with fewer accidents (errors).

This breakthrough makes it much easier to build smaller, faster, and more secure communication systems, quantum computers, and advanced imaging tools.

1. Problem Statement

The rapid advancement of high-dimensional optical encoding (e.g., for spatial-division multiplexing, quantum key distribution, and super-resolution imaging) has created a critical need for efficient spatial mode sorters. These devices must separate a set of $M$ orthogonal spatial modes (such as Hermite-Gaussian, Laguerre-Gaussian, or Bessel-Gaussian) from a single input channel into $M$ distinct output channels with minimal cross-talk.

Current Limitations:

Complexity: Traditional high-performance sorters (e.g., Multi-Plane Light Conversion or MPLC) typically require multiple cascaded optical layers (often 4 to 10+), increasing system bulk, alignment sensitivity, and calibration complexity.
Restrictions: Existing single-layer solutions, such as the Fork grating or Log-Polar transformations, are often limited to specific mode families (e.g., Orbital Angular Momentum with zero radial index) or suffer from high cross-talk.
Performance Trade-offs: There is a lack of a unified, single-plane device capable of sorting arbitrary mode families with near-zero cross-talk while maintaining a predictable power transmission efficiency.

2. Methodology

The authors propose a novel single-plane spatial mode sorter based on an analytic derivation using Fourier optics principles.

Theoretical Framework:
- The sorter utilizes a single mask $S(x, y)$ placed in the input plane.
- The mask is defined as a superposition of the complex conjugates of the target modes, each modulated by a linear phase term to steer the light to specific detector locations $(\alpha_m, \beta_m)$ in the far field:
  $S(x, y) = \frac{1}{\sqrt{M}} \sum_{m'=1}^{M} f^*_{m'}(x, y) e^{i \frac{2\pi}{\lambda z} (\alpha_{m'} x + \beta_{m'} y)}$
- Mechanism: When an input mode $f_m$ passes through the mask, the term $f_m \cdot f^*_m$ creates a constructive interference peak at the designated detector location $(\alpha_m, \beta_m)$ . Cross-terms ( $f_m \cdot f^*_{m'}$ where $m \neq m'$ ) result in off-axis scattering that is filtered out or falls between detectors, provided the detectors are spaced by a distance greater than the modes' bandwidth $\Omega$ .
Optimality Proof: The authors provide a mathematical proof demonstrating that for any random arrangement of detector positions, this single-layer configuration achieves the optimal power transmission possible for a single layer. They show that the power transmission coefficient scales as $O(M^{-1})$ , which is the theoretical lower bound for such a system.
Experimental Implementation:
- Setup: A 632.8 nm He-Ne laser beam is expanded and encoded with specific spatial modes using a phase-only Spatial Light Modulator (SLM-A).
- Sorting: The beam passes through a 4f system to a second SLM (SLM-B) acting as the sorter mask. A subsequent 2f system projects the far-field pattern onto a camera.
- Phase-Only Approximation: Although the theoretical mask requires both amplitude and phase modulation, the authors demonstrate that a phase-only approximation (using only the phase of the complex mask) yields nearly identical performance for the tested modes, making it compatible with standard phase-only SLMs.

3. Key Contributions

Universal Single-Plane Sorter: Development of a device capable of sorting diverse mode families (Hermite-Gaussian, Laguerre-Gaussian, Bessel-Gaussian) and mixed sets with a single optical element.
Generalization of Fork Gratings: The authors prove that their general derivation reduces to the well-known Fork grating configuration when applied to Orbital Angular Momentum (OAM) modes with zero radial index, unifying previous specific methods under a broader theoretical framework.
Infinite Mode Limit: An analytical derivation for the limit $M \to \infty$ , providing closed-form expressions for sorters that can theoretically sort all modes of a family simultaneously.
Mode Generation: By operating the device in reverse (illuminating the sorter with a Gaussian beam), the system functions as an efficient spatial mode generator, producing multiple distinct modes simultaneously.
Spectroscopic Application: The wavelength sensitivity of the sorter (where sorting angle scales with wavelength) is exploited to create a compact spectroscopic technique for measuring wavelength shifts.

4. Results

Sorting Performance:
- Cross-talk: For orthogonal mode sets, the experimental cross-talk was extremely low, averaging 2.7%, with sorting efficiencies (diagonal elements) averaging 96.6%.
- Mode Families: Successful sorting was demonstrated for HG, LG, and BG modes, as well as mixed sets.
- MUBs: When sorting Mutually Unbiased Bases (non-orthogonal sets), cross-talk increased as predicted by the inner products of the modes, validating the theoretical model.
Power Transmission:
- Experimental data confirmed the theoretical scaling of power transmission as $O(M^{-1})$ . For $M=4$ , transmission was high; as $M$ increased, transmission decreased inversely, consistent with the optimality proof.
Robustness:
- Phase Noise: The sorter showed high robustness to random phase noise (up to $\sigma \approx 0.5\pi$ ), as the noise distributes incoherently rather than coherently redirecting energy to wrong detectors.
- Wavelength Sensitivity: While the sorter is robust to small wavelength shifts, large shifts cause significant displacement of the sorting spots. This was quantified and used for spectroscopy.
Spectroscopy: Using Bessel-Gaussian modes, the team demonstrated the ability to estimate wavelength shifts ( $\Delta \lambda$ ) based on pixel displacement ( $d$ ) with errors as low as 1.1% for shifts up to 37 nm.

5. Significance and Impact

Architectural Simplicity: This work offers a drastic reduction in system complexity compared to MPLC, replacing bulky multi-layer cascades with a single plane. This is crucial for compact, integrated photonic systems.
Signal Fidelity: The method achieves near-zero cross-talk for orthogonal modes, outperforming geometric transformation methods (like Log-Polar) which typically suffer from -10 to -20 dB cross-talk due to approximations.
Scalability vs. Efficiency: While the $O(M^{-1})$ power loss is a limitation for very large $M$ , the authors argue that for many applications (e.g., QKD, classical multiplexing), this loss is manageable via amplification or is negligible compared to channel attenuation.
Versatility: The ability to function as both a sorter and a generator, and to handle mixed mode families, makes this a versatile tool for quantum information processing, high-capacity communications, and advanced imaging.
Future Outlook: The paper suggests that while currently realized via SLMs, the design can be fabricated as a fixed, compact device using direct laser printing, paving the way for commercial deployment in quantum and classical optical networks.