Ablation Depth & Threshold

There are two parts. The first measures the beam radius and threshold fluence from your own crater diameters using the Liu D² method. The second uses that threshold to estimate depth per pulse, incubation over many pulses, crater volume, and the fluence that removes the most material per joule.

1 · Threshold from crater diameters (Liu method)

Measured craters

Paste from a spreadsheet: tab-, comma-, or space-separated. Lines starting with # are ignored. Use at least three energies spanning a factor of 5–10 or more.

Fit results

Threshold fluence (peak)Fth—
Threshold pulse energyEth—
Beam radius at the surface (1/e²)w₀—
Spot diameter (1/e²)2w₀—
Slope of D² vs ln E2w₀²—
Coefficient of determinationR²—
Points used—

D² versus pulse energy

2 · Depth, incubation & removal efficiency

Process

A fitted parameter, not the optical absorption depth. Get it from the slope of measured depth per pulse against ln F₀.

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S = 1 means no incubation. Metals commonly show S ≈ 0.8–0.9. For scanned processes, use Neff from the overlap tool.

Results

Depth per pulse after N pulsesδ ln(F₀/Fth(N))—
Single-pulse depthδ ln(F₀/Fth(1))—
Estimated total depth after N pulses—
Incubated thresholdFth(N)—
Fluence ratioF₀/Fth(N)—
Removal efficiencyV/Ep—
Optimum peak fluencee²·Fth(N)—
Efficiency vs. optimum—
Crater diameterD—
Volume removed per pulse—

Removal efficiency vs. fluence

Threshold from crater diameters

A Gaussian beam ablates wherever the local fluence \(F_0e^{-2r^2/w_0^2}\) exceeds the threshold. The crater diameter therefore depends on pulse energy as

\[ D^2 = 2w_0^2 \ln\!\left(\frac{E_p}{E_\text{th}}\right), \qquad F_\text{th} = \frac{2E_\text{th}}{\pi w_0^2}. \]

Plotting \(D^2\) against \(\ln E_p\) gives a straight line. Its slope is \(2w_0^2\) and its intercept with \(D^2 = 0\) is \(E_\text{th}\) (Liu, 1982). You measure the beam radius at the work surface and the threshold together, without a separate beam profiler. A bent line suggests a non-Gaussian beam, a second ablation regime, or craters measured at different pulse numbers.

Craters made with N pulses each give \(F_\text{th}(N)\), not the single-pulse threshold. Repeating the fit at several N and plotting \(\ln F_\text{th}(N)\) against \(\ln N\) gives \(S - 1\) as the slope.

Logarithmic depth law

If the absorbed energy density decays exponentially into the material with a characteristic length \(\delta_\text{eff}\), material is removed down to the depth where the deposited energy density falls to its threshold:

\[ L = \delta_\text{eff}\,\ln\!\left(\frac{F_0}{F_\text{th}}\right). \]

Depth grows with the logarithm of fluence. Doubling the fluence adds a fixed increment \(\delta_\text{eff}\ln 2\) instead of doubling the depth. For ultrashort pulses on metals, two logarithmic regimes are commonly seen (Nolte et al., 1997). At low fluence \(\delta_\text{eff}\) is close to the optical penetration depth, of order 10 nm. At higher fluence a larger electron heat-diffusion length applies, of order 100 nm.

δeff is a fit parameter, not 1/α. Take crystalline silicon at 1064 nm. Absorption is weak (α of order 10 cm⁻¹ at room temperature), so the optical penetration depth is about a millimetre. Nanosecond ablation depths there are set by heating, melting, and the rise in absorption with temperature, not by optical absorption. Measure δeff for your material, wavelength, and pulse duration. Do not take it from optical constants.

Incubation

Repeated exposure lowers the threshold through accumulated defects, roughness, and oxidation. The widely used empirical model of Jee et al. is

\[ F_\text{th}(N) = F_\text{th}(1)\,N^{\,S-1}, \]

where \(S = 1\) means no incubation. The total-depth estimate here sums the logarithmic law pulse by pulse, with the \(n\)-th pulse seeing \(F_\text{th}(n)\). It suits shallow features. In deep grooves and holes, the walls spread the beam, the effective fluence at the bottom drops, and the depth saturates.

Maximum removal efficiency at e² × threshold

Integrating the logarithmic depth profile over a Gaussian spot gives the volume per pulse and the volume per unit energy:

\[ V = \frac{\pi \delta_\text{eff} w_0^2}{4}\ln^2\!\frac{F_0}{F_\text{th}}, \qquad \eta = \frac{V}{E_p} = \frac{\delta_\text{eff}}{2F_0}\ln^2\!\frac{F_0}{F_\text{th}}. \]

\(\eta\) is maximized at \(F_0 = e^2F_\text{th} \approx 7.39\,F_\text{th}\), where \(\eta_\text{max} = 2\delta_\text{eff}/(e^2F_\text{th})\) (Neuenschwander et al., 2010). The peak is broad: anywhere from about 3× to 25× threshold keeps at least 75 % of \(\eta_\text{max}\). Running far above it wastes energy as heat and plasma. When average power is the limit, the efficient strategy is to raise the repetition rate or split the beam, not to raise the fluence.

Worked example

Take F₀ = 1.5 J/cm², Fth(1) = 0.2 J/cm², δeff = 20 nm, N = 10, S = 0.85, and a 30 µm spot. These are illustrative values.

Assumptions and limits

References

  1. J. M. Liu, “Simple technique for measurements of pulsed Gaussian-beam spot sizes,” Opt. Lett. 7, 196–198 (1982).
  2. Y. Jee, M. F. Becker, and R. M. Walser, “Laser-induced damage on single-crystal metal surfaces,” J. Opt. Soc. Am. B 5, 648–659 (1988).
  3. S. Nolte, C. Momma, H. Jacobs, A. Tünnermann, B. N. Chichkov, B. Wellegehausen, and H. Welling, “Ablation of metals by ultrashort laser pulses,” J. Opt. Soc. Am. B 14, 2716–2722 (1997).
  4. B. Neuenschwander et al., “Processing of metals and dielectric materials with ps-laser pulses: results, strategies, limitations and needs,” Proc. SPIE 7584, 75840R (2010).
  5. D. Bäuerle, Laser Processing and Chemistry, 4th ed., Springer (2011).

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