How it is calculated
Heat released at a surface spreads with the diffusion equation. After a time \(t\), the temperature profile from an instantaneous planar source falls as \(\exp(-x^2/4Dt)\), so it drops to 1/e at
where \(k\) is the thermal conductivity, \(\rho\) the density, and \(c_p\) the specific heat. Authors differ on the prefactor: \(\sqrt{D\tau}\), \(\sqrt{2D\tau}\), and \(2\sqrt{D\tau}\) all appear. Both common conventions are shown, so check which one a source uses before comparing numbers. Diffusion length scales only with \(\sqrt{\tau}\). Going from 10 ns to 10 ps shrinks it by a factor of about 32, not 1000.
Thermal confinement
If \(L_\text{th} \lesssim \delta\), the optical penetration depth, heat stays in the volume where it was absorbed during the pulse. The pulse is thermally confined. Energy density then builds up efficiently and ablation is relatively clean, with a small heat-affected zone. If \(L_\text{th} \gg \delta\), heat leaks into the bulk during the pulse. More energy is needed to reach ablation, and melt and recast appear. For metals, with \(\delta \approx\) 10–30 nm, confinement requires picosecond or shorter pulses.
Heat accumulation between pulses
After each pulse, heat must spread out of the irradiated spot before the next pulse arrives. Spreading over the beam radius takes about
Above roughly \(f_\text{acc}\), residual heat from earlier pulses raises the baseline temperature. The process then behaves more like a quasi-CW heat source: the heat-affected zone grows, melt and burr formation increase, and the effective threshold drops. Using the full diameter instead of the radius gives a 4× lower, more conservative onset. The transition is gradual because residual heat decays slowly, so treat \(f_\text{acc}\) as an order-of-magnitude guide. Pulse overlap raises the number of pulses per point and makes accumulation worse (see the overlap tool).
Material properties
| Material | k (W/m·K) | ρ (kg/m³) | cp (J/kg·K) | D (mm²/s) |
|---|---|---|---|---|
| Copper | 401 | 8933 | 385 | 117 |
| Aluminum (pure) | 237 | 2702 | 903 | 97.1 |
| Silicon | 148 | 2330 | 712 | 89.2 |
| Stainless steel 304 | 14.9 | 7900 | 477 | 3.95 |
| Ti-6Al-4V | 6.7 | 4430 | 526 | 2.88 |
| Fused silica | 1.38 | 2203 | 740 | 0.85 |
| Borosilicate (Borofloat 33) | 1.2 | 2230 | 830 | 0.65 |
| Soda-lime glass | 1.0 | 2500 | 750 | 0.53 |
| Water (25 °C) | 0.607 | 997 | 4181 | 0.146 |
| PMMA | 0.19 | 1190 | 1420 | 0.11 |
These are typical room-temperature values. Alloys, glasses, and polymers vary with composition and supplier, and every material's diffusivity changes with temperature. Most metals' diffusivity falls as they heat toward melting. Use measured data for your material when accuracy matters.
Worked example
Stainless steel 304 (D = 3.95 mm²/s) with 10 ns pulses: \(L_\text{th} = 2\sqrt{3.95\ \text{mm}^2/\text{s}\times 10\ \text{ns}} \approx 0.40\ \mu\text{m}\). That is about 20× the optical penetration depth, so the process is not thermally confined. The same steel with 10 ps pulses gives 12.6 nm, which is comparable to δ. For a 30 µm spot, \(t_w = (15\ \mu\text{m})^2/(4D) = 14\ \mu\text{s}\), so heat accumulation sets in around 70 kHz. At 100 kHz with significant overlap, expect accumulation. In fused silica the same spot accumulates heat from about 15 kHz. In copper the onset is about 2 MHz.
Assumptions and limits
- Classical (Fourier) diffusion with constant properties. Phase changes, latent heat, and temperature-dependent properties are not included.
- For metals at pulse durations below a few picoseconds, electrons and the lattice are not in equilibrium. Heat initially spreads by hot-electron transport, which the two-temperature model describes. The classical \(L_\text{th}\) then underestimates the early heat penetration.
- The accumulation onset compares single-spot timescales. Scanned processes also depend on overlap, line spacing, and part geometry. Thin sheets and small features accumulate heat faster than a semi-infinite solid.
References
- T. L. Bergman, A. S. Lavine, F. P. Incropera, and D. P. DeWitt, Fundamentals of Heat and Mass Transfer, 7th ed., Wiley (2011), Appendix A (property data).
- H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed., Oxford University Press (1959).
- D. Bäuerle, Laser Processing and Chemistry, 4th ed., Springer (2011).