A 16-emitter phase-locked array. Sixteen independent light sources, each with its own slightly-detuned frequency, coupled together until they fall into a single coherent voice — and the scattered glow collapses into one bright, narrow beam.
Kuramoto coupled-phase-oscillator model · the real math behind laser-array mutual phase-locking
Each emitter i carries a phase θi(t) and a natural frequency ωi. The phases evolve by the Kuramoto equation — each oscillator is pulled toward the others in proportion to the sine of their phase difference:
The collective state is the complex order parameter — the average of every emitter written as a point on the unit circle:
Here r ∈ [0,1] measures coherence. When the emitters are scattered, their unit vectors cancel and r ≈ 0 (incoherent). When they lock, the vectors align and r → 1. For ω drawn from a symmetric unimodal distribution with half-width set by the spread γ, mean-field theory gives a critical coupling below which no synchrony is possible and above which r grows continuously from zero:
where g(0) is the peak of the natural-frequency density. Sweep K through Kₜ on the slider above and watch the transition: the phase dots on the unit circle gather from a diffuse ring into a tight clump, r climbs off the floor, and the far-field beam on the right sharpens.
The right panel is the coherent far-field intensity of the array — the emitters treated as a line of point sources radiating at angle φ. The field amplitude is the same order-parameter sum, now evaluated per direction, and intensity is its magnitude squared:
When phases are random (r ≈ 0), the sum is an incoherent walk: peak intensity scales like N and the light spills across a broad, dim lobe. When phases lock (r → 1), the terms add in phase in the forward direction: peak intensity scales like N² and the energy concentrates into a bright, narrow main lobe with array side-lobes. That N → N² brightness gain, bought purely by coherence, is the whole point of a phase-locked emitter array.
real physicsThe Kuramoto model of coupled phase oscillators (Y. Kuramoto, 1975) is established, heavily-studied dynamics. The equation above, the complex order parameter r·e^{iψ}, and the incoherent→locked transition at a critical coupling Kₜ are correctly implemented here and integrated in real time by 4th-order Runge–Kutta. Mutual phase-locking of real laser arrays — semiconductor diode bars, fiber-laser arrays, VCSEL arrays — is genuinely described by Kuramoto-type coupling, and the N→N² coherent far-field brightening is standard array optics.
design / disclosureThe specific “16-emitter chromatic laser” as a named device is a ROOT0 design disclosure — a concept for how one might arrange and couple such an array. It is not a built laser, not a measured instrument, and not a granted patent. No performance number on this page was measured on hardware.
symbolic toyThe far-field panel uses a simplified 1-D point-source model (uniform spacing, scalar field, no gain medium, no cavity dynamics, no thermal or mode effects). The word “chromatic” is aspirational framing for a multi-frequency array; the simulation locks phases, not colors, and does not model actual wavelength conversion. Read the beam panel as an illustration of the coherence principle, not a device spec.
To disclose a design is real work; it is not a granted patent or a fabricated device.
Y. Kuramoto, “Self-entrainment of a population of coupled non-linear oscillators,” Int. Symp. on Mathematical Problems in Theoretical Physics, Lecture Notes in Physics 39, 420–422 (1975).
S. H. Strogatz, “From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators,” Physica D 143, 1–20 (2000).
J. A. Acebrón et al., “The Kuramoto model: A simple paradigm for synchronization phenomena,” Rev. Mod. Phys. 77, 137 (2005).