How it is calculated
A Gaussian beam with 1/e² radius \(w_0\) and pulse energy \(E_p\) has the fluence distribution
The peak irradiance follows from the peak power, \(I_0 = 2P_\text{peak}/(\pi w_0^2)\), where \(P_\text{peak} = kE_p/\tau\) with \(k = 1\), 0.9394, or 0.8814 for rectangular, Gaussian, or sech² pulses (see pulse energy & peak power). For a top-hat beam of diameter \(d\), the fluence is uniform: \(F = E_p/(\pi d^2/4)\).
Peak or average? The factor of two
Two conventions are in common use, and they differ by exactly a factor of two for a Gaussian beam:
- Peak (on-axis) fluence \(F_0 = 2E_p/(\pi w_0^2)\). This is the value at the center of the beam. Ablation thresholds, damage thresholds (ISO 21254 uses the effective area \(A_\text{eff} = \pi w_0^2/2\)), and most process literature use it.
- Average fluence \(E_p/(\pi w_0^2)\): the energy divided by the area of the 1/e² circle. It is a common shortcut, but it equals \(F_0/2\), and only 86.5 % of the energy actually falls inside that circle.
Comparing an average fluence to a threshold quoted as a peak value overstates the process margin by a factor of two. Before comparing a process window against literature or a vendor specification, check which convention each source uses.
Threshold diameter
Where \(F(r)\) exceeds a threshold \(F_\text{th}\), the material is modified or ablated. Setting \(F(r) = F_\text{th}\) gives
The feature grows only logarithmically with fluence. Doubling the pulse energy at \(F_0/F_\text{th} = 5\) widens it by only about 20 %. Fitting D² against ln Ep is the Liu method for measuring \(w_0\) and \(F_\text{th}\), implemented in the ablation depth & threshold tool.
Worked example
A 20 µJ, 10 ns Gaussian pulse focused to a 30 µm (1/e²) spot:
The average over the 1/e² circle is 2.83 J/cm². With a 1 J/cm² threshold, the modified diameter is 27.9 µm, and 82 % of the pulse energy lands inside it.
Assumptions and limits
- The spot diameter is the 1/e² diameter at the work surface. Defocus increases it: see Gaussian beam propagation.
- Fluence is incident fluence. Absorbed fluence is lower by the surface reflectance, which for metals in the near-IR is roughly 60–98 %.
- A real top-hat has soft edges, and a real Gaussian may have M² > 1 or ellipticity. Measure the profile (ISO 11146) when accuracy matters.
- At high irradiance, plasma shielding and nonlinear absorption change how much energy reaches the surface. Those effects are not modeled.
References
- J. M. Liu, “Simple technique for measurements of pulsed Gaussian-beam spot sizes,” Opt. Lett. 7, 196–198 (1982).
- ISO 21254-1:2011, Lasers and laser-related equipment — Test methods for laser-induced damage threshold — Part 1: Definitions and general principles.
- W. M. Steen and J. Mazumder, Laser Material Processing, 4th ed., Springer (2010).