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Non-uniform modal power distribution caused by disorder in multimode fibers

This paper demonstrates through four converging experimental and numerical approaches that disorder in multimode fibers drives modal crosstalk toward steady states with non-uniform power distributions favoring lower-order modes, which are accurately described by a weighted Bose-Einstein law.

Original authors: Mario Zitelli

Published 2026-01-27

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Mario Zitelli

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

The Big Picture: The "Crowded Highway" Problem

Imagine a multimode fiber optic cable not as a single glass strand, but as a multi-lane highway with many different lanes (modes) running side-by-side. In an ideal world, if you send a car (light signal) into Lane 1, it stays in Lane 1. If you send cars into all lanes equally, they all arrive at the destination with the same amount of traffic.

However, real highways aren't perfect. They have potholes, bumps, and slight curves (disorder). In a fiber optic cable, these imperfections cause cars to randomly switch lanes. This is called Random Mode Coupling (RMC).

The main discovery of this paper is that even if you start with an equal number of cars in every lane, the random lane-switching caused by the road's imperfections eventually leads to a very uneven traffic jam. By the time the cars reach the end, the "inner lanes" (low-order modes) are packed with traffic, while the "outer lanes" (high-order modes) are almost empty.

The Four Ways They Tested This

The researchers didn't just guess; they used four different methods to prove this happens, and all four methods told the same story:

The Computer Simulation (The "Digital Twin"): They built a complex math model on a computer that mimics how light waves wiggle and interact as they travel through a bumpy fiber. They programmed the fiber to have random imperfections.
The "Traffic Flow" Model: Instead of tracking individual waves, they used a simpler model that just tracks the total "amount of power" (like total traffic volume) moving between groups of lanes. This model assumes power flows more easily from outer lanes to inner lanes than the other way around.
The Real-World Lab Test (Classical Light): They sent actual laser pulses (like fast-moving cars) through 5 kilometers of real fiber optic cable. They used special equipment to inject light equally into different "lanes" and measured what came out the other side.
The Single-Photon Test (The "Ghost Car" Experiment): To be absolutely sure this wasn't a weird effect of many light waves crashing into each other, they sent one single photon (a single particle of light) at a time. Even with just one "ghost car" at a time, the same pattern emerged: the photon was more likely to end up in the inner lanes than the outer ones.

The Surprising Result: The "Weighted Bose-Einstein" Law

The researchers found that this uneven distribution isn't random chaos; it follows a specific mathematical rule called a Weighted Bose-Einstein (wBE) law.

The Analogy:
Imagine a crowded party where people are dancing.

The Disorder: The floor is slightly uneven, causing people to stumble and bump into each other.
The Result: Even if everyone starts dancing in a circle with equal energy, the bumps eventually cause everyone to drift toward the center of the room (the low-order modes). The people on the edges (high-order modes) get pushed out or lose their energy.

The paper shows that the fiber naturally "prefers" the inner lanes. It's not because the inner lanes are better, but because the physics of the random bumps makes it statistically harder to stay in the outer lanes.

What About "Loss"?

You might think, "Maybe the outer lanes just have more holes in them, so the light leaks out?" The researchers checked this carefully. They measured how much light was lost due to the fiber's imperfections (Mode-Dependent Loss).

They found that while light loss does make the outer lanes emptier, it is not the main cause. Even if you mathematically remove the "leakage" from the equation, the uneven distribution remains. The random switching itself is enough to create the imbalance.

The Conclusion

The paper concludes that in a long enough fiber optic cable, disorder creates order.

If you send light in equally, the natural imperfections of the glass will eventually sort the light so that the "inner" modes get all the power, and the "outer" modes get very little. This happens whether you are sending a massive burst of laser light or just a single photon.

This finding is important because it proves that the fiber itself has a "memory" or a preferred state (steady state) that it naturally drifts toward, described by that specific math law (wBE), regardless of how you start the journey.

Technical Summary: Non-uniform Modal Power Distribution Caused by Disorder in Multimode Fibers

Problem Statement
Multimode fibers (MMFs) and multicore fibers (MCFs) are critical for increasing optical transmission capacity via space-division multiplexing (SDM). However, their performance is limited by transmission impairments, specifically random-mode coupling (RMC) caused by fiber disorder (micro/macro bending, radius/index variations) and mode-dependent loss (MDL). While RMC causes a random exchange of power between ortho-normal modes, previous numerical simulations yielded conflicting results regarding the steady-state power distribution. Some models suggested a nearly uniform distribution, while others indicated non-uniformity depending on the specific disorder model used. Furthermore, it remained unclear whether the promotion of lower-order modes (LOMs) at the expense of higher-order modes (HOMs) was solely a result of MDL or if RMC alone could generate such non-uniform steady states.

Methodology
To resolve these discrepancies and investigate the steady-state modal power distribution, the authors employed four distinct experimental and numerical approaches over several kilometers of graded-index (GRIN) multimode fiber:

Coupled Generalized Nonlinear Schrödinger Equations (GNLSE): Numerical simulations modified to include RMC effects via a field coupling model. Disorder was modeled using random refractive index perturbations ( $\Delta n$ ) with uniform distribution, calculating coupling coefficients based on overlap integrals of transverse fields.
Power-Flow Equations (PFE): A numerical model describing the bi-directional diffusive flow of power between modal groups. This model treats power exchange between adjacent non-degenerate groups, neglecting fast intra-group coupling.
Classical Linear Regime Experiment: Transmission of 1 Gb/s, low-power pulses (negligible Kerr and Raman nonlinearities) over 5 km of OM4 GRIN fiber. Input modes were injected and output modes decomposed using multi-plane light conversion (MPLC) technology. Modal power was measured and normalized to isolate the effects of RMC from material loss and insertion loss.
Quantum Regime Experiment: Similar to the classical experiment but utilizing single-photon pulses and photon-counting techniques. This approach tested whether the non-uniform distribution persists in the ergodic case of single-photon exchanges, confirming the statistical nature of the phenomenon.

In all cases, Mode-Dependent Loss (MDL) was characterized and isolated to measure the specific contribution of RMC.

Key Results
All four approaches converged on the same conclusion: fiber disorder alone (RMC) is capable of generating steady states characterized by non-uniform modal power distributions.

Direction of Power Flow: The steady state is characterized by the promotion of lower-order modes (LOMs) at the expense of higher-order modes (HOMs). This occurs even when equal power is injected into all input modes and even in the absence of MDL.
Mathematical Description: The resulting non-uniform distributions are properly described by a weighted Bose-Einstein (wBE) law. The experimental and simulation data fitted the wBE model with high accuracy (R-squared > 0.95).
Comparison with Other Models: The Rayleigh-Jeans (RJ) law, often used for nonlinear optical scrambling, failed to properly describe the distribution of LOMs in the linear RMC-dominated regime.
Quantum Ergodicity: The quantum experiment demonstrated that the non-uniform distribution is an ergodic property of the system. Similar distributions were obtained whether measuring multi-photon optical power or counting repeated single-photon events, confirming that the phenomenon is intrinsic to the mode coupling statistics rather than a classical intensity effect.
Role of MDL: While MDL enhances the preferential flow toward LOMs, the experiments confirmed that RMC alone is sufficient to generate the non-uniform steady state.

Significance and Claims
The paper claims that the non-uniform modal power distribution observed in long multimode fibers is a fundamental consequence of random-mode coupling, not merely an artifact of fiber imperfections or loss. The findings establish that:

Steady State Formation: Sufficiently long fibers (hundreds of meters to kilometers) naturally evolve toward a steady state where power concentrates in lower-order modes, regardless of the input distribution.
Thermodynamic Analogy: The validity of the weighted Bose-Einstein law suggests that the multimode fiber system behaves like a thermodynamic system reaching maximum Boltzmann entropy, where the "temperature" and "chemical potential" are defined by optical parameters.
Quantum-Classical Correspondence: The confirmation of these statistics in the single-photon regime bridges classical optical thermodynamics and quantum optics, paving the way for future quantum thermodynamics experiments in optical multimodal systems.

The authors emphasize that these results resolve previous conflicts in the literature by demonstrating that the specific modeling of disorder (whether via random potentials or matrix products) consistently leads to the same wBE-distributed steady state when RMC is the dominant mechanism.