Model note · Inside the calculated voxel

How the two-photon lithography model works

The lab is a mechanistic sketch: a vectorial focus drives a spatial reaction–diffusion model, then a bath-accessible development pass removes weakly converted material.

Start with the light Read the papers

The field shape changes with NA and wavelength before chemistry begins. The bright core is sampled over each simulation voxel.

01 / Scope

A model for reasoning, not a virtual instrument certificate

Every displayed field comes from the Rust/Wasm simulation. The renderer does not paint a desired outcome, but the model is not calibrated to a named commercial resin or printer.

The controls separate measured process inputs—power, pulse duration, repetition rate, speed, NA, wavelength—from dimensionless chemistry coefficients. Literature supports the mechanisms and qualitative dependencies; it does not make the default coefficients experimentally identified. Use the lab to ask causal questions, then validate any quantitative prediction against a real resin, objective, and development protocol.

Input Literature-shaped mechanism Exploratory coefficient

02 / Volume

One occupancy, one evolving resin volume

The Micro‑Benchy is voxelized once. Occupied cells define the intended material; the solver tracks photoinitiator, oxygen, radical activity, conversion, and remaining mass in the full 3D domain. Off-target cells can still activate when the optical kernel or diffusing species reach them.

That distinction is why the amber material can disagree with the slate target. The target is geometry. The amber volume is a calculated consequence.

03 / Scan path

Geometry becomes a timed exposure history

Layer height, hatch spacing, hatch angle, contour count, pass count, and scan speed build a deterministic path through the occupancy. Moving between adjacent path samples adds illuminated dwell time; longer jumps add dark time, during which diffusion, oxygen recovery, and radical loss continue.

Power and speed therefore do different things. Power changes the source amplitude quadratically. Speed changes how long the source dwells along the same geometry. Repeated passes revisit the path after the intervening chemistry has evolved.

04 / Optics + power

Where the power is calculated

The power control is specimen-plane average power. It is not divided uniformly among voxels.

Pulse energyEₚ = P / f

Peak-power proxyP̂ = Eₚ / τ

Vector fieldE(r) = Debye(NA, λ, polarization)

PI spectrumA(λ) = exp[−4 ln 2 ((λ−λₚᵢ)/160 nm)²]

Local sources(r) ∝ A(λ) · |E(r)|⁴ · P² / (f τ)

The objective semi-angle is derived from NA and a fixed immersion refractive index. A circularly polarized vectorial Debye integral produces the focal electric field, following the high-aperture foundation of Richards and Wolf [1]. The pupil is normalized to fixed total power, so raising NA concentrates the same specimen power instead of silently adding energy.

Wavelength also changes initiation through an exploratory Gaussian photoinitiator response. Its peak is adjustable and its fixed 160 nm width is intentionally only a qualitative stand-in until a measured two-photon absorption spectrum is supplied for a named material.

Two-photon initiation follows the squared local intensity, consistent with the focal confinement demonstrated in early 3D two-photon polymerization work [2–4]. The current implementation voxel-averages the narrow central lobe when the grid would otherwise under-resolve it. Its absolute dose constant is anchored to the default 16 mW, 80 MHz, 100 fs, NA 1.4, 780 nm case. That anchor is numerical, not a claim of resin-specific calibration.

Read this carefully

“16 mW” is a real process-style input. The conversion predicted from it remains relative until the two-photon cross-section, initiator quantum yield, optical losses, and resin kinetics are fitted to experiment.

05 / Chemistry

A deliberately compact reaction–diffusion system

Exposure depletes photoinitiator and generates radicals. Oxygen inhibits generation and quenches radicals; photoinitiator, oxygen, and radicals diffuse; radicals also disappear through linear loss and bimolecular termination. Surviving radicals accumulate conversion. These coupled effects are grounded in measured oxygen inhibition and spatial photopolymerization models[5–6].

Initiator∂ₜp = Dₚ∇²p − βsp

Radicals∂ₜr = Dᵣ∇²r + ηsp − (δ + qo)r − κr²

Oxygen∂ₜo = Dₒ∇²o − χqor

Conversion∂ₜx = γr(1 − x)

These equations preserve the causal vocabulary of radical photopolymerization without attempting a full resin reaction network. The coefficients identified as exploratory in the article are stable dimensionless controls, not values copied from one paper.

06 / Development

The bath attacks what it can reach

Development begins at solvent-connected surfaces, not at the rectangular simulation boundary. A distance transform estimates bath depth through the occupied material. Local mass then decays faster where conversion is below the gel point and slower where a stronger network resists dissolution.

Remaining mass∂ₜm = −kᵈcᵈ(depth)m / exp(ρx)

Real post-processing also includes solvent choice, rinsing, shrinkage, adhesion, drying, and capillary collapse. Development studies show that these steps can decide whether delicate TPP structures survive [7]; the lab currently models dissolution only, not mechanics or drying.

07 / Limits

What the simulation leaves out

Objective transmission, aberration, interface mismatch, and measured beam profile.
A resin-specific two-photon cross-section and fitted kinetic rate constants.
Temperature rise, viscosity change, autoacceleration, vitrification, and shrinkage.
Developer transport coupled to moving boundaries, swelling, stress, adhesion, and capillary forces.
Arbitrary uploaded STL slicing—the current run remains tied to the bundled benchmark occupancy.

Those omissions are boundaries, not footnotes. They define what a counterfactual run can teach and what still needs an experiment.

08 / Papers