Pump–probe · Gaussian beams
Both spots are Gaussian and concentric, so the probe does not sample one fluence — it samples a distribution. The calculator also accounts for the angle of incidence: a tilted beam is stretched along the plane of incidence by 1/cos θ, which lowers the fluence accordingly.
Enter either width — the other converts automatically (D1/e² ≈ 1.699 × FWHM). Spot sizes are measured perpendicular to propagation; angles from the sample normal, both beams in the same plane of incidence.
Peak fluence F₀
at beam center
Probe-wtd avg F̄
quote this value
Average / peak
Cut through both spots on the sample surface (each normalized to its own peak).
Probe-intensity-weighted density of pump fluence, p(F) — normalized to its maximum.
Subscript p = pump, s = probe. All beams TEM₀₀ Gaussian, centers coincident, plane of incidence along x.
A Gaussian fluence profile with peak F0 and 1/e² radius w:
At r = w the fluence has fallen to 1/e² ≈ 13.5 % of the peak. The two diameter conventions:
A beam of 1/e² radius w hitting the sample at angle θ from the normal is stretched along the plane of incidence; the footprint is an elliptical Gaussian:
The same pulse energy is spread over a 1/cos θ larger area, so every fluence below picks up exactly one factor of cos θp.
Integrating the footprint over the surface must return the pulse energy E:
Note the factor 2: the peak is twice the "energy ÷ 1/e² area" estimate. In practical units (E in µJ, Dp in µm):
A linear signal averages the pump fluence over the probe intensity Is. With elliptical Gaussians the average separates per axis:
At normal incidence kx = ky = k and this reduces to the familiar
Handy checkpoints at normal incidence:
| Ds = Dp/3 | F̄/F0 = 90 % |
| Ds = Dp/2 | F̄/F0 = 80 % |
| Ds = Dp | F̄/F0 = 50 % |
Changing variables from position to fluence, the probe-intensity-weighted density of pump fluence on [0, F0] is
where I0 is the modified Bessel function (the page evaluates it numerically). At normal incidence B = 0 and this collapses to the pure power law
whose mean reproduces §4:
For k > 1 (probe smaller than pump) the weight is concentrated near the peak fluence — the central fluence dominates the signal. For k < 1 the low-fluence wings dominate and the density diverges as F → 0.
• Keep Ds ≤ Dp/3 if you want the "uniform fluence" approximation to hold within 10 %.
• For a signal that scales nonlinearly as Fn, the weighted moment is
— averaging distorts nonlinear signals even more strongly than linear ones.
• Both angles are measured from the sample normal and both beams are assumed to lie in the same plane of incidence (the usual pump–probe geometry).
• Assumes the probe is weak (non-perturbing) and both footprints stay concentric; lateral walk-off lowers the effective fluence further.