Ultrafast pump–probe · two Gaussian beams in vacuum

Pump–Probe Focal Overlap

The pump and probe cross inside the sample, and the signal grows with how much pump the probe photons actually pass through. This page integrates that intensity-weighted overlap over space and time. It reports fv, which reads 100% when the pump sits at its best position and delay.

Focal overlap fv
–%
100% = best pump position and delay at this crossing angle
Absolute overlap η
–

probe-weighted pump intensity ÷ peak

Pump intensity seen by probe
–

η × I0

Crossing-angle factor
–

best overlap ÷ collinear best

Crossing plane

Top view (x–z) through the probe axis, across the full sample depth. Drag to move the pump focus.

Pump Probe Signal (pump × probe)

Where the signal comes from

Beam cross-section

The x–y plane at one depth, looking along the probe. The pump spot is stretched along x by 1/cos θ. Drag to move the pump.

Overlap map

fv for every pump position. Click or drag to place the pump there.

0%100%· contours at 50% and 90%

Delay scan

fv versus pump–probe delay with the pump where it is now. Positive delay: the probe arrives after the pump.

Position scan

fv as the pump moves along one axis through its current position.

AxisFWHM≥ 90% of peakSlope here

How fv is calculated

Linear-response pump–probe signal of two TEM00 Gaussian beams with Gaussian, transform-limited pulses in vacuum.

Beams

The probe travels along z and focuses at the origin. The pump travels in the x–z plane at the crossing angle θ, with its focus at (Δx, Δy, Δz). Each beam, in its own coordinates (transverse ρ, longitudinal s):

I(ρ, s, t) = I₀ · (w₀/w(s))² · exp(−2ρ²/w(s)²) · exp(−4 ln2 · (t − s/c)²/τ²)
w(s) = w₀ √(1 + (s/zR)²), zR = π w₀²/λ

Pulse fronts are perpendicular to each beam. When the beams cross at θ, the arrival-time difference changes by sin θ / c per µm across x. This smears the cross-correlation and is why a large angle shortens the useful overlap.

Signal

S = ∫sample d³r ∫ dt · D(r, t) · Iprobe(r, t)
  • 2PA: D = Ipump. Nondegenerate pump + probe photon absorption, present only while the pulses overlap.
  • TA: D = N ∝ ∫−∞t Ipump(t′) e−(t−t′)/T₁ dt′. The pump leaves an excited population that the probe reads later.
  • 2P-excited TA: as TA, with Ipump² as the source. The effective pump spot is √2 narrower.

The two overlap numbers

fv = S / max S (max over pump position and delay, same angle)
η = S / (Dpeak · L · Eprobe) = ⟨D⟩probe / Dpeak

fv tells you how well aligned you are, from 0 to 100%. η is the absolute factor you need when converting a measured ΔA into a cross-section: the probe sees an effective pump intensity η·I₀ (2PA) or fluence η·F₀ (TA). The crossing-angle factor compares the best overlap at θ with the best collinear overlap of the same beams.

Check value. For collinear beams, no diffraction, at zero delay (2PA):

η = wpump² / (wpump² + wprobe²) · τpump / √(τpump² + τprobe²)

Numerics and assumptions

  • At each depth z the x, y and t integrals are done analytically. This includes the pulse-front-tilt smear and the excited-state rise and decay. The z integral uses an adaptive trapezoid grid. A brute-force quadrature agrees to about 10⁻⁴ for typical cases.
  • Small-signal limit. There is no pump depletion, saturation or ground-state bleaching of the pump, and the probe absorption stays linear.
  • Vacuum propagation. There is no refraction, group-velocity walk-off or dispersion, and polarization and orientation effects are ignored.
  • Intensity sets η·I₀ and the drawing brightness. It does not change fv or η, because the signal is linear in each beam.