Two-Photon Absorption Calculator

The TPA cross section delta expresses how efficiently a molecule absorbs two photons simultaneously. One GM (1 x 10^-50 cm^4 s photon^-1) represents the product of two one-photon cross-sectional areas (each roughly the molecular size in cm^2) and a temporal coincidence window of about 10^-17 s. A larger value means the molecule is more likely to undergo TPA per unit photon flux squared. Typical organic dyes have cross sections of a few to a few hundred GM, while specially designed push-pull chromophores and metal nanoclusters can reach thousands or even hundreds of thousands of GM.

TPA uses two photons each carrying half the energy needed for the electronic transition. For molecules that absorb in the UV or visible range (300-500 nm), the TPA excitation wavelength falls in the NIR (600-1000 nm). NIR light scatters and is absorbed far less by biological tissue than UV or visible light, so it penetrates much more deeply. This is the reason two-photon microscopy can image intact tissue at depths of hundreds of micrometres, whereas conventional confocal microscopy is limited to roughly 50-100 μm before scattering degrades the image.

TPA rate scales with the peak intensity squared, not the average intensity. A pulsed laser concentrates the same average power into very short bursts, dramatically increasing the peak intensity. For a laser with repetition rate f and pulse width tau_p delivering average power P_avg, the peak power is approximately P_avg / (f x tau_p). Ultra-short pulses (femtoseconds) therefore give orders-of-magnitude more TPA events than CW illumination at the same average power, which is why mode-locked femtosecond Ti:Sapphire lasers are the standard TPA excitation source.

The TPA cross section delta (GM) describes the absorption probability. The two-photon action cross section sigma_2p = delta x phi_f, where phi_f is the fluorescence quantum yield, describes how much detectable fluorescence is produced per excitation event. Since fluorescence quantum yields range from near zero to nearly one, the action cross section is always less than or equal to the TPA cross section. When comparing fluorophores for two-photon imaging, the action cross section is the more practical figure because it tells you directly how much signal to expect.

The two most common methods are Z-scan and two-photon excited fluorescence (TPEF) comparison. In a Z-scan, the sample is translated through the focal point of a Gaussian beam and the transmitted intensity is recorded; the nonlinear absorption appears as an on-axis dip when the sample is at focus. In TPEF comparison, the fluorescence signal from the unknown sample is compared with that from a reference fluorophore with a known cross section (e.g. fluorescein at 782 nm, delta approximately 36 GM). Both methods require careful characterisation of the beam parameters, particularly the beam radius at the focus.

For one-photon absorption, intensity falls exponentially with path length: I(x) = I_0 exp(-alpha x). For TPA, the governing equation is I(x) = I_0 / (1 + beta x I_0), where beta is the bulk TPA coefficient in cm/W. This hyperbolic rather than exponential decay means TPA attenuation is negligible at low intensity and becomes significant only when beta x I_0 x x is comparable to 1. For dilute solutions of organic fluorophores, the bulk beta is typically so small that the sample is nearly transparent at all practical intensities, and TPA is useful as a microscopy/lithography tool rather than an attenuator.