Pump–probe · Gaussian beams
Calculates incident fluence for concentric Gaussian pump and probe spots. Oblique incidence stretches each footprint along the common plane of incidence, and the probe-weighted value describes a signal that is linear in local pump fluence and probe intensity.
Enter either width; the other converts automatically (D1/e² ≈ 1.699 × FWHM). Diameters are measured perpendicular to propagation. Angles are measured from the sample normal.
Incident peak F₀
at pump-beam center
Incident effective F̄
linear probe-weighted value
Effective / peak
Model scope: these are incident, per-pulse fluences—not absorbed fluence or deposited energy density. The effective value assumes concentric TEM₀₀ Gaussian spots, aligned footprint axes, a weak probe, and a detected signal proportional to FpumpIprobe. Apply independently known reflection, transmission, absorption-depth, aperture, or nonlinear-response corrections separately.
One-dimensional center cut along the plane of incidence. Pump and probe curves are normalized to their own peaks; the shaded product is the local contribution to a linear probe signal, also normalized to its center value.
Probe-intensity-weighted probability density of local pump fluence, p(F). The vertical scale is divided by its maximum, so it shows relative density—not probability or signal fraction at a single fluence.
Subscript p = pump, s = probe. The beams are TEM₀₀ Gaussians with coincident centers and aligned sample-plane ellipse axes; x lies in the shared plane of incidence.
A circular Gaussian fluence profile with incident peak F0 and 1/e² radius w is
At r = w, the fluence is 1/e² ≈ 13.5% of the peak. The two diameter conventions are
The input diameters are defined in a plane perpendicular to the direction of beam propagation.
A circular beam incident at angle θ from the sample normal is stretched along the plane of incidence. Its sample-plane 1/e² radii become
The footprint is therefore an elliptical Gaussian:
The calculator applies this projection independently to the pump and probe. It assumes that their ellipse axes are aligned.
Integrating the pump fluence over the sample surface must recover the pulse energy E:
At normal incidence this reduces to
This is incident fluence per pulse. Reflection, transmission and absorption are not included in F0.
For a weak probe whose detected signal is proportional to local probe intensity, the normalized spatial sampling weight is
If the sample response is linear in local pump fluence, the measured effective incident fluence is
For concentric, aligned elliptical Gaussians, the integrals separate along the two axes and give
The result approaches the peak when the probe is much smaller than the pump along both axes.
The red curve is normalized pump fluence, and the blue curve is normalized probe intensity. The shaded center cut is
This product is the normalized local contribution to a linear probe signal. It is not the sampling weight by itself: the sampling weight is Ws, which is proportional to probe intensity alone.
The plotted cut is one-dimensional; the effective fluence is evaluated from the full two-dimensional Gaussian footprints.
The probe samples a distribution of local incident pump fluences. Its normalized probability density is defined by
For the aligned elliptical Gaussian model, let z = F/F0. The exact density is
Here I0 is the modified Bessel function of the first kind. When the pump and probe footprints have the same aspect ratio— including normal incidence and the collinear equal-angle case—kx = ky = k, so the density becomes
The chart displays p(F)/max[p(F)], not the absolute density. Its vertical coordinate therefore describes relative density shape rather than probability contained at one exact fluence.
For a nonlinear, saturated or thresholded material response, average the actual response function rather than replacing the distribution by a single fluence:
If the detection process scales as a higher power of probe intensity, replace Is by the appropriate detection weighting.
A simple absorbed-fluence correction can only be applied when independently measured reflected and transmitted fractions are meaningful:
Depth-dependent absorption, multilayer interference, non-Gaussian beam maps, clipping, hot spots, lateral pump–probe offset, rotated ellipses, finite apertures, spatially varying time delay and uncertainty propagation require a more complete model.