Increasing Pump-Probe Signal toward Asymptotic Limits

Abstract

Optimization of pump-probe signal requires a complete understanding of how signal scales with experimental factors. In simple systems, signal scales quadratically with molar absorptivity, and linearly with fluence, concentration, and path length. In practice, scaling factors weaken beyond certain thresholds (e.g., OD > 0.1) due to asymptotic limits related to optical density, fluence and path length. While computational models can accurately account for subdued scaling, quantitative explanations often appear quite technical in the literature. This Perspective aims to present a simpler understanding of the subject with concise formulas for estimating absolute magnitudes of signal under both ordinary and asymptotic scaling conditions. This formulation may be more appealing for spectroscopists seeking rough estimates of signal or relative comparisons. We identify scaling dependencies of signal with respect to experimental parameters and discuss applications for improving signal under broad conditions. We also review other signal enhancement methods, such as local-oscillator attenuation and plasmonic enhancement, and discuss respective benefits and challenges regarding asymptotic limits that signal cannot exceed.

Figure 1

Characteristic measures of a Gaussian beam, including spot size , beam waist , Rayleigh length , and aperture diameter .

Figure 2

Plot of pump fluence as a function of the propagation depth into the sample in cases of low and high optical density.

Figure 3

Comparison of ΔOD scaling for linear and competitive absorbers. (A) Scaling with respect to OD as a function of concentration or path length. (B) Scaling with respect to OD as a function of analyte molar absorptivity.

Figure 4

Comparison of ΔOD for the competitive absorber model computed by approximate and exact solutions to Beer’s law, as a function of optical density, for varying ratios of bandwidth to line width.

Figure 5

Plot of absorption versus resonant fluence for a saturable absorber measured by an ultrafast pulse.

Figure 6

Pump–probe signal as a function of resonant fluence for a linear absorber and a saturable absorber. OD(0) is the linear absorbance measured in the weak fluence limit.

Figure 7

Contour plot of signal versus optical density and fluence for the saturable competitive absorber.

Figure 8

Typical photon conversion efficiency measured in near-IR BBO and mid-IR AgGaS₂.

Figure 9

Plot illustrating the effective path length of the excitation volume relative to the Rayleigh length.

Figure 10

Plot of ΔOD versus path length for various solvent absorptivities. Asterisks identify points at which the path length is equal to the inverse solvent absorptivity.

Figure 11

Four models of pump–probe signal under varying conditions of relative resonant fluence and total optical density. Quantities necessary to evaluate these formulas include the analyte optical density (OD_a), total optical density (OD_T), the fraction of pump fluence resonant with the analyte line shape (F̃), and the saturation fluence of the analyte (F_sat).