STUDY #28 · 2026 · IN OBSERVATION

Talbot Carpet

A model-driven visual study of grating self-imaging.

MOVING IMAGE — ONE FULL TALBOT PERIOD slit f 0.22 · M 96 · chroma 0.12 · ζ +1 · exact loop · plate

WHAT IS THIS

In 1836 H. F. Talbot noticed that behind an illuminated diffraction grating, the grating's own image reappears at regular distances — with no lens, no mirror, nothing but propagation. Lord Rayleigh derived the revival distance, z_T = 2a²/λ. The full intensity map between revivals — position × height — is the Talbot carpet.

The model is the paraxial diffraction-order sum: each order m carries a phase that grows like m² × height, which is why arithmetic — not optics — decides what the light does. This study evaluates that sum directly on the GPU, in real time, at every pixel.

the loom — the full carpet, grating plane to first revival slit f 0.22 · M 96 · chroma 0.12 · ζ 0→1 · plate

Motif Talbot effect / near-field diffraction / fractional revivals / number theory in light

Method A small simulator was generated and modified with AI assistance, then ported to a real-time GPU (GLSL) renderer. The visual output was selected through parameter exploration.

Observation At rational heights p/q the field locks into q shifted copies of the grating (Gauss-sum weights, verified to machine precision); at irrational heights it stays lace — the golden-ratio profile measured a fractal box dimension of 1.43 (theory 3/2). The carpet mirrors at half the Talbot distance and repeats exactly at one.

Reference H. F. Talbot, "Facts relating to optical science. No. IV," Philosophical Magazine (Third Series), vol.9, 401-407 (1836); M. V. Berry & S. Klein, "Integer, fractional and fractal Talbot effects," Journal of Modern Optics, vol.43, 2139-2164 (1996).

Tools Python / NumPy / three.js / React / GLSL / ffmpeg / AI coding assistant

Year 2026

This is not a scientific simulation result, but a visual interpretation of the phenomenon.

SAME FAMILY, THE NEAR FIELD

Interference draws fringes where waves cross. The near field re-draws the grating itself.

Study #06 — Wave Interference Study #28 — Talbot Carpet

The regime far field — fringes settle far from the sources near field — right behind the grating, before the orders separate

The canvas a plane of space the waves cross position × height — the propagation axis itself is in the picture

What decides the pattern geometry — path differences between sources arithmetic — the denominator q of the height ζ = p/q

The image fringes — the sources' own shape is never rebuilt the grating rebuilds itself — no lens, exact, at every z_T

PARAMETERS EXPLORED

param meaning effect on the image

ζ = z/z_T height above the grating — the one dial (z_T = 2a²/λ) rational p/q locks the field into q shifted copies; irrational heights never resolve — the golden ratio weaves the deepest lace

f slit fill factor — the thread's thickness wide slits print a white fan; narrow slits invert the carpet into shadow spires

M kept diffraction orders — the loom's thread count 2 weaves a soft cosine twill, 16 crystallises into lace; past 32 the eye saturates (sinc envelope)

grating amplitude / binary phase / blazed a phase grating is invisible at ζ = 0 and develops itself downstream; blazed π exposes the glide symmetry as diagonal chevrons

chroma source bandwidth — laser to white light real dispersion, z_T ∝ 1/λ per channel: red revives sooner, blue later, and the carpet braids into rainbow

Each image below records its exact parameter set.

THE MATHEMATICS the model behind the images

One period of a grating written as diffraction orders — the loom's threads — each propagated with a phase quadratic in the order number. Everything in the carpet follows from that exponent.

U(x,\zeta) \;=\; \sum_{m=-M}^{M} c_m\, e^{2\pi i m x}\, e^{-2\pi i m^2 \zeta}, \qquad I = |U|^2

The paraxial mode sum in grating units (a = 1), with ζ = z/z_T and z_T = 2a²/λ — Rayleigh's distance. The phase is quadratic in the order m, so what the light does at height ζ is a question about m²ζ mod 1: arithmetic, not optics.

U\!\left(x,\tfrac{p}{q}\right) = \sum_{s=0}^{q-1} B_s\, t\!\left(x - \tfrac{s}{q}\right), \qquad B_s = \frac{1}{q} \sum_{n \bmod q} e^{-2\pi i\, p n^2/q}\, e^{2\pi i n s/q}

The fractional Talbot effect (Berry & Klein 1996): at rational heights the field is exactly q shifted copies of the grating, weighted by Gauss sums. Verified here to 1.9 × 10⁻¹³, with Σ|B_s|² = 1 and |B_s| = 1/√5 exact at q = 5.

U\!\left(x+\tfrac{1}{2},\;\zeta+\tfrac{1}{2}\right) = U(x,\zeta), \qquad I(x,\zeta) = I(x,1-\zeta)

The carpet's weave is its symmetry: a glide — half a period across, half a Talbot distance up — that holds for any grating (4.3 × 10⁻¹⁴, checked on a blazed profile), and a mirror at half the Talbot distance for symmetric amplitude gratings (2.8 × 10⁻¹³).

A visual interpretation of the model, not a claim of scientific precision. The engine is paraxial (a/λ ≳ 50); the non-paraxial walk-off was measured to fall as (λ/a)² — slope 2.000 — and the GPU field matches a float64 reference at six heights to Pearson 1.0000000.

SELECTED STILLS — 6

the half-revival band — hourglass cusps where the grating returns shifted by half a period slit f 0.24 · M 96 · ζ 0.40–0.60 · brocade

golden lace — the most irrational height, where the image never resolves slit f 0.18 · M 128 · ζ φ±0.06 · atomlaser

the pillars — a π phase grating, invisible at the grating plane, developing itself downstream phase π · f 0.5 · M 96 · ζ 0–0.5 · xray

the chevron — a blazed grating exposes the carpet's glide symmetry as diagonal weave blazed π · M 96 · ζ 0→1 · ember

the shadow spires — narrow slits invert the white-light fan; the bars' shadows fray into rainbow slit f 0.13 · M 128 · chroma 1 · ζ 0.001–0.04 · spectral

PROCESS — PARAMETER SWEEPS

The spine of the exploration — one grating held fixed while the height crosses the rational ladder: ζ = 1, 1/2, 1/3, 1/4, 2/5, 3/7 and the golden ratio, each height read as carpet, profile and zoom strip. At every rational the profile locks into q copies; at the golden ratio it refuses. Four more sheets swept fill factor (the hero band lives at f 0.16–0.28), kept orders (the eye saturates near M ≈ 32), grating types — a blazed 2π grating is a single order and the carpet vanishes: nothing to interfere with, nothing woven — and real RGB dispersion.

the rational ladder — carpets, profiles and zoom strips across seven heights slit f 0.24 · M 128 · ζ ∈ {1, 1/2, 1/3, 1/4, 2/5, 3/7, φ} · zoom strips ±0.03 · plate

SIGNATURE — THE LOOM OF RATIONALS

Rationals resolve, irrationals weave: the light reads the denominator of its height.

At any rational height ζ = p/q the paraxial field is exactly q shifted copies of the grating, weighted by Gauss sums — quadratic exponential sums straight out of number theory. The denominator counts the copies: small q prints bold images, large q splits the light into fine ribbed lace. The engine was checked against that identity before any image was kept — the reconstruction agrees to 1.9 × 10⁻¹³, the weights carry unit energy, and at q = 5 every |B_s| = 1/√5 exactly.

Between the rationals the image never resolves. At the golden ratio — the most irrational height — Berry and Klein proved the intensity profile is a fractal curve; this engine measures its box dimension at 1.426 against their 3/2 (the convergence is logarithmically slow), while the smooth rational cut at ζ = 1/2 measures 1.043. The weave itself is symmetry: the carpet mirrors at half the Talbot distance and glides by half a period across, half a distance up — both identities verified near machine precision.

golden lace under the box-counting rule — the profile that never resolves box dimension 1.426 (theory 3/2) · ζ = 1/2 → 1.043 · |B_s| = 1/√5 at q = 5

COLOUR = REAL IMAGE-FORMING MEDIA

The monochrome palettes ground to real image-forming media: a lantern plate — indigo glass, amber lace; X-ray film — Talbot–Lau interferometers really do print these fringes into film, dense where light lands, clear where it cancels; and the green fluorescence of cold-atom matter-wave Talbot experiments. Only the brocade palette — indigo ground, gold thread — is a declared artistic choice: the weave metaphor itself.

The polychrome images are white-light Talbot: the three RGB channels are propagated separately, each with its own Talbot distance (z_T = 2a²/λ at 650 / 550 / 460 nm), so red revives sooner and blue later — the rainbow braiding is the physics of dispersion, not a grade.

white light propagated as three real wavelengths — the braid is dispersion, not grading slit f 0.13 · chroma 1 · z_T ∝ 1/λ · 650/550/460 nm · spectral

All palettes are artistic approximations of these media, not measurements.

REFERENCES

H. F. Talbot. "Facts relating to optical science. No. IV." The London and Edinburgh Philosophical Magazine and Journal of Science (Third Series), vol.9, no.56, 401-407 (1836).
Lord Rayleigh. "On copying diffraction-gratings, and on some phenomena connected therewith." Philosophical Magazine (Fifth Series), vol.11, no.67, 196-205 (1881).
M. V. Berry, S. Klein. "Integer, fractional and fractal Talbot effects." Journal of Modern Optics, vol.43, no.10, 2139-2164 (1996).
J. Wen, Y. Zhang, M. Xiao. "The Talbot effect: recent advances in classical optics, nonlinear optics, and quantum optics." Advances in Optics and Photonics, vol.5, no.1, 83-130 (2013).

INTERACTIVE STUDY

A grating lit from below weaves a carpet of light with no lens — and the carpet is fully analytic, so the only motion is yours. THE dial is height ζ: at rational heights — 1/2, 1/3, 2/5 — the pattern clicks into q crisp copies and the lock label lights (a Ford-circle rule, window ~1/q²); between them it dissolves into lace, and the golden height never resolves at all. The left strip is the whole carpet — click it to jump; drag the right pane to scrub; the bright midline is what a screen at your height would record. It is a deliberately simplified instrument, a 40-order window with a few curated knobs and no export, separate from the full engine used to author the finished works.

SIMPLIFIED INSTRUMENTPARAXIAL MODE SUM · 40 ORDERS · ζ LIVE

This interactive study is not intended as a scientifically validated reproduction. It is a visual interpretation generated from an implemented model and curated parameter exploration — and it is a deliberately simplified instrument, separate from the full engine used to author the finished works.