pith. sign in

arxiv: 2604.04903 · v1 · submitted 2026-04-06 · ⚛️ physics.optics · physics.comp-ph

Maximally localized modes of a multimode fiber

Nicolas Barr\'e This is my paper

Pith reviewed 2026-05-10 19:28 UTC · model grok-4.3

classification ⚛️ physics.optics physics.comp-ph
keywords multimode fibermaximally localized modesWannier functionsspatial spread minimizationgraded-index fibermode basis optimizationphotonic lanternconcentric ring modes
0
0 comments X

The pith

Minimizing the total spatial spread of modes in a multimode fiber produces self-organized concentric rings without any geometric constraints imposed.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a numerical method that searches for the most compact orthonormal basis of fiber modes by applying unitary transformations to an initial set and minimizing the sum of each mode's spatial spread. When applied to the Laguerre-Gaussian modes of a graded-index fiber, the optimization yields solutions in which the modes arrange themselves into concentric rings for every tested count from 6 to 55. Spot sizes and ellipticities change from ring to ring in patterns that standard geometric packing cannot forecast. At larger mode numbers the rings themselves become irregular, with neither the number of spots per ring nor their internal arrangement following obvious symmetry, showing that fully symmetric configurations cease to be minima of the spread functional. A constrained version of the same optimizer can force any desired bundle shape and then report the extra spread cost incurred.

Core claim

The optimization procedure identifies the optical equivalent of maximally localized Wannier functions by finding the unitary transformation of the Laguerre-Gaussian basis that minimizes the sum of individual mode spreads; the resulting modes spontaneously form concentric rings whose radii, spot sizes, and ellipticities evolve systematically yet non-geometrically, and for high mode counts the arrangements within rings lose regularity.

What carries the argument

The spread functional minimized over all unitary transformations of a fixed orthonormal mode basis, which selects the linear combinations that are most concentrated in space.

If this is right

  • Modes form concentric rings spontaneously for all tested counts between 6 and 55.
  • Spot sizes and ellipticities vary between rings in ways geometric packing cannot predict.
  • For large mode counts neither the number of spots per ring nor the arrangement inside a ring remains regular.
  • A constrained optimizer can enforce any prescribed bundle geometry and directly quantify the resulting increase in total spread.
  • The approach supplies a concrete, physically motivated cost for any target mode layout in photonic lantern design.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The ring structures may appear in other wave systems where a similar spread functional can be defined, such as acoustic or quantum waveguides.
  • Comparing the optimized spreads against known analytic bounds could test whether the irregular patterns found at high mode counts are truly global minima.
  • The method offers a way to rank candidate mode layouts for applications by their localization cost rather than by ad-hoc symmetry assumptions.

Load-bearing premise

That the numerical minimization reliably reaches the global minimum of the spread functional and that the Laguerre-Gaussian starting basis does not itself force the observed ring organization.

What would settle it

Running the same minimization starting from a different complete orthonormal basis such as Hermite-Gaussian modes and obtaining a lower total spread with a non-ring arrangement for mode counts above 30.

Figures

Figures reproduced from arXiv: 2604.04903 by Nicolas Barr\'e.

Figure 1
Figure 1. Figure 1: MLFM obtained by minimizing Ω over the LG basis of a graded-index fiber, for 𝑁 = 6 to 55. Each spot is represented by its intensity centroid (dot) and an ellipse whose semi-axes are proportional to 𝜎𝑟 and 𝜎𝜃 (scaled by 0.45 for clarity). The two families of solutions (𝑛1 = 1 and 𝑛1 = 3) are clearly visible. bundle geometries generated from the recurrence rule, with 𝑛1 = 3 for 𝑁 = 36 and 𝑁 = 55, and 𝑛1 = 1 … view at source ↗
Figure 2
Figure 2. Figure 2: MLFM obtained with constrained optimization (𝛽 = 1) targeting the regular concentric ring geometry, for 𝑁 = 36, 45, and 55. The total spread is within 0.8%, 1.4%, and 1.7% of the unconstrained optimum respectively, confirming that the regular packing is near-optimal. The constrained optimization successfully recovers symmetric solutions in all three cases. Interestingly, the centroids of the resulting mode… view at source ↗
Figure 3
Figure 3. Figure 3: Intensity (top) and phase (bottom) of representative MLFM for the 𝑁 = 36 constrained solution. Columns 1–2: two adjacent spots from ring 3; columns 3–4: two adjacent spots from ring 4. Within each ring, the dominant intensity lobes are nearly identical while the nodal line patterns differ, illustrating the quasi-symmetry of the constrained solutions. Modes from different rings have partially overlapping sp… view at source ↗
read the original abstract

This article presents an optimization method to find the most spatially concentrated basis of a multimode fiber, obtained by minimizing the sum of the spatial spreads of the individual modes over all unitary transformations of a given orthonormal mode set. The resulting modes are the optical analogue of maximally localized Wannier functions in solid-state physics. We apply the method to the Laguerre-Gaussian basis of a graded-index fiber for mode counts ranging from 6 to 55. In all cases, the modes spontaneously organize into concentric rings without any geometric constraint being imposed. The spot sizes and ellipticities evolve from one ring to the next in ways that geometric packing approaches cannot predict. For large mode counts, the optimizer finds solutions where neither the number of spots per ring nor the spots within a given ring follow a regular pattern, indicating that the fully symmetric arrangement is no longer a minimum of the spread functional. A constrained variant of the method enables the optimizer to target any prescribed bundle geometry while quantifying its localization cost, opening a route to physically grounded photonic lantern design.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces a numerical optimization method to construct maximally localized modes in a multimode graded-index fiber. The approach minimizes the sum of individual mode spatial spreads over all unitary transformations of an initial Laguerre-Gaussian orthonormal basis. For mode counts from 6 to 55 the resulting modes are reported to self-organize into concentric rings without imposed geometric constraints; spot sizes and ellipticities vary across rings, and for larger counts the optimizer yields irregular (non-periodic) arrangements within rings. A constrained variant of the optimizer is also presented that can target prescribed bundle geometries while reporting the associated localization penalty. The construction is positioned as the optical analogue of maximally localized Wannier functions.

Significance. If the reported configurations are global minima of the spread functional, the work supplies a concrete, parameter-free route to localized fiber modes and a quantitative tool for assessing the localization cost of arbitrary photonic-lantern geometries. The spontaneous emergence of ring structures and the breakdown of regularity at high mode counts would constitute a nontrivial observation about the landscape of the spread functional on the unitary group.

major comments (3)
  1. [Results] Results (paragraph beginning 'For large mode counts...'): The claim that 'the fully symmetric arrangement is no longer a minimum of the spread functional' for N ≳ 30 requires a direct numerical comparison of the achieved spread value against the spread obtained under a symmetry-constrained unitary. No such comparison, nor the numerical value of the minimized spread itself, is supplied; without it the irregular solutions cannot be asserted to be globally preferred.
  2. [Method] Method (description of the minimization): The unitary manifold for N = 55 is 55²-dimensional. The manuscript provides no information on the optimizer (gradient descent, manifold optimization, etc.), step-size schedule, convergence tolerance, or number of random initial unitaries tested. Multiple random starts and comparison against the symmetric subspace are necessary to substantiate that the reported irregular patterns are not local minima.
  3. [Abstract and Results] Abstract and Results: No quantitative metrics (final spread values, improvement relative to the LG basis, error bars from multiple runs) are reported for any of the 6–55 mode cases. The central qualitative claims therefore rest on unverified numerical outcomes.
minor comments (2)
  1. [Method] The definition of the spread functional (sum of second moments) should be written explicitly with the precise normalization and integration domain used in the fiber cross-section.
  2. [Figures] Figure captions should state the exact mode count N and the initial basis for each panel so that the progression from regular to irregular rings can be traced quantitatively.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and constructive feedback on our manuscript. The comments highlight important gaps in quantitative support and methodological detail that we will address in a revised version. Below we respond point by point to the major comments.

read point-by-point responses

  1. Referee: [Results] Results (paragraph beginning 'For large mode counts...'): The claim that 'the fully symmetric arrangement is no longer a minimum of the spread functional' for N ≳ 30 requires a direct numerical comparison of the achieved spread value against the spread obtained under a symmetry-constrained unitary. No such comparison, nor the numerical value of the minimized spread itself, is supplied; without it the irregular solutions cannot be asserted to be globally preferred.

    Authors: We agree that a direct numerical comparison is required to substantiate the claim. The manuscript currently presents only the qualitative observation of irregular patterns without reporting the spread values or comparing them to symmetry-constrained optimizations. In the revision we will add these comparisons, including the numerical values of the minimized spread functional for both the unconstrained optimizer and the symmetry-constrained case, for N around 30 and above. This will provide quantitative evidence that the irregular configurations achieve lower total spread. revision: yes

  2. Referee: [Method] Method (description of the minimization): The unitary manifold for N = 55 is 55²-dimensional. The manuscript provides no information on the optimizer (gradient descent, manifold optimization, etc.), step-size schedule, convergence tolerance, or number of random initial unitaries tested. Multiple random starts and comparison against the symmetric subspace are necessary to substantiate that the reported irregular patterns are not local minima.

    Authors: The referee is correct that the manuscript omits essential details on the numerical procedure. We will expand the Methods section to specify the optimization algorithm (a Riemannian manifold optimizer on the unitary group), the step-size schedule, convergence tolerance, and the number of random initial unitaries tested. We will also report the spread values obtained from multiple independent runs and compare them explicitly to the spread achieved within the symmetric subspace, thereby addressing the concern that the irregular patterns might be local rather than global minima. revision: yes

  3. Referee: [Abstract and Results] Abstract and Results: No quantitative metrics (final spread values, improvement relative to the LG basis, error bars from multiple runs) are reported for any of the 6–55 mode cases. The central qualitative claims therefore rest on unverified numerical outcomes.

    Authors: We acknowledge that the absence of quantitative metrics weakens the presentation. Although the manuscript emphasizes the spontaneous organization into rings, we will add a table (or supplementary figure) in the revised Results section that reports the final spread values, the relative improvement over the original Laguerre-Gaussian basis, and standard deviations or ranges obtained from multiple optimization runs for each mode count from 6 to 55. These additions will place the qualitative observations on a firmer numerical footing. revision: yes

Circularity Check

0 steps flagged

No circularity: direct numerical minimization of explicitly defined spread functional

full rationale

The paper defines the spread functional explicitly as the sum of individual mode spreads and computes its minimum over the unitary group acting on the Laguerre-Gaussian basis. The reported ring organization and irregular patterns are direct numerical outputs of this optimization procedure for N=6 to 55; they are not presupposed by any equation, fitted parameter, or self-citation. No load-bearing step reduces the result to a definition or prior self-referential claim. The method is self-contained against external benchmarks (Wannier-function analogy is descriptive, not used to derive the numerical solutions).

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on the domain assumption that spatial spread is a physically meaningful localization metric and that unitary transformations preserve the mode orthonormality required for the fiber.

axioms (1)
  • domain assumption Spatial spread is an appropriate objective function for quantifying mode localization in fibers
    Directly used as the quantity to be minimized over unitary transformations.

pith-pipeline@v0.9.0 · 5467 in / 1244 out tokens · 56437 ms · 2026-05-10T19:28:16.856131+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

  1. [1]

    T. A. Birks, I. Gris-Sánchez, S. Yerolatsitis, S. G. Leon-Saval, and R. R. Thomson. The photonic lantern. Advances in Optics and Photonics, 7(2):107–167, Jun 2015.doi:10.1364/AOP.7.000107

    work page doi:10.1364/aop.7.000107 2015

  2. [2]

    Velázquez-Benítez, J

    Amado M. Velázquez-Benítez, J. Enrique Antonio-López, Juan C. Alvarado-Zacarías, Nicolas K. Fontaine, RolandRyf,HaoshuoChen,JuanHernández-Cordero,PierreSillard,ChigoOkonkwo,SergioG.Leon-Saval, andRodrigoAmezcua-Correa. Scalingphotoniclanternsforspace-divisionmultiplexing.ScientificReports, 8(1):8897, Jun 2018.doi:10.1038/s41598-018-27072-2

    work page doi:10.1038/s41598-018-27072-2 2018

  3. [3]

    Nemanja Jovanovic, Pradip Gatkine, Narsireddy Anugu, Rodrigo Amezcua-Correa, Ritoban Basu Thakur, et al. 2023 astrophotonics roadmap: pathways to realizing multi-functional integrated astrophotonic instruments.Journal of Physics: Photonics, 5(4):042501, oct 2023.doi:10.1088/2515-7647/ace869

    work page doi:10.1088/2515-7647/ace869 2023

  4. [4]

    Geometricrequirements for photonic lanterns in space division multiplexing.Optics Express, 20(24):27123–27132, Nov 2012

    NicolasK.Fontaine,RolandRyf,JossBland-Hawthorn,andSergioG.Leon-Saval. Geometricrequirements for photonic lanterns in space division multiplexing.Optics Express, 20(24):27123–27132, Nov 2012. doi:10.1364/OE.20.027123

    work page doi:10.1364/oe.20.027123 2012

  5. [5]

    Davenport, Momen Diab, Kalaga Madhav, and Martin M

    John J. Davenport, Momen Diab, Kalaga Madhav, and Martin M. Roth. Optimal smf packing in photonic lanterns: comparing theoretical topology to practical packing arrangements.J. Opt. Soc. Am. B, 38(7):A7– A14, Jul 2021.doi:10.1364/JOSAB.413640

    work page doi:10.1364/josab.413640 2021

  6. [6]

    Graham, B.D

    R.L. Graham, B.D. Lubachevsky, K.J. Nurmela, and P.R.J. Östergård. Dense packings of congruent circles in a circle.Discrete Mathematics, 181(1):139–154, 1998.doi:10.1016/S0012-365X(97)00050-2

    work page doi:10.1016/s0012-365x(97)00050-2 1998

  7. [7]

    Gregory H. Wannier. The structure of electronic excitation levels in insulating crystals.Physical Review, 52:191–197, Aug 1937.doi:10.1103/PhysRev.52.191. 6

    work page doi:10.1103/physrev.52.191 1937

  8. [8]

    Physical Review B , author =

    Nicola Marzari and David Vanderbilt. Maximally localized generalized wannier functions for composite energy bands.Physical Review B, 56:12847–12865, Nov 1997. URL:https://link.aps.org/doi/10. 1103/PhysRevB.56.12847,doi:10.1103/PhysRevB.56.12847

    work page doi:10.1103/physrevb.56.12847 1997

  9. [9]

    Marzari , author A

    Nicola Marzari, Arash A. Mostofi, Jonathan R. Yates, Ivo Souza, and David Vanderbilt. Maximally localized wannier functions: Theory and applications.Reviews of Modern Physics, 84:1419–1475, Oct 2012.doi:10.1103/RevModPhys.84.1419

    work page doi:10.1103/revmodphys.84.1419 2012

  10. [10]

    Journal of Open Source Software , keywords =

    Nicolas Barré. Fluxoptics.jl: A differentiable wave optics framework for inverse design in julia.Journal of Open Source Software, 11(118):9734, 2026.doi:10.21105/joss.09734

    work page doi:10.21105/joss.09734 2026

  11. [11]

    A method for solving the convex programming problem with convergence rate𝑜(1/𝑘 2)

    Yurii Nesterov. A method for solving the convex programming problem with convergence rate𝑜(1/𝑘 2). In Doklady Akademii Nauk SSSR, volume 269, page 543, 1983. 7

    work page 1983