Two-Photon Absorption Calculator

Two-photon absorption (TPA) is a nonlinear optical process in which an atom or molecule simultaneously absorbs two photons, transitioning from its ground state to a higher excited state via a virtual intermediate state. The combined energy of both photons must match the energy gap between the two states. TPA was first theoretically predicted by Maria Göppert-Mayer in 1931.

This calculator uses the equation N = ½ · δ · ϕ² · τ, where N is the number of two-photon excitations per molecule per pulse, δ is the two-photon absorption cross-section (in GM units), ϕ is the photon flux at the focal point (photons/cm²/s), and τ is the pulse duration. The factor of ½ arises from the statistical nature of simultaneous two-photon absorption.

Photon absorption is the process by which a photon transfers its energy to an atom, ion, or molecule, promoting it to a higher energy state. In linear (one-photon) absorption, one photon is absorbed per event. In two-photon absorption, two photons are absorbed simultaneously, requiring much higher optical intensities due to its nonlinear nature.

The Göppert-Mayer (GM) unit is the standard unit for the two-photon absorption cross-section δ. One GM equals 10⁻⁵⁰ cm⁴·s per photon. It is named in honor of Maria Göppert-Mayer, who theoretically described the TPA process. Typical molecular cross-sections range from 1 to thousands of GM, while specially designed TPA dyes can exceed 10,000 GM.

TPA is nonlinear because the absorption rate depends on the square of the optical intensity rather than the intensity itself. This means doubling the laser power quadruples the TPA rate. Significant TPA only occurs at very high photon densities, which is why pulsed lasers with tight focusing are typically used in TPA experiments.

The TPA cross-section is commonly measured using two-photon excited fluorescence (TPEF), where the sample is excited by a focused pulsed laser and the resulting fluorescence intensity (which scales as I²) is compared to a reference compound with a known cross-section. Z-scan and pump-probe techniques are also widely used for direct TPA cross-section measurements.

TPA has numerous applications including two-photon fluorescence microscopy (allowing 3D imaging deep within biological tissues), two-photon photodynamic therapy for cancer treatment, 3D optical data storage, microfabrication via two-photon polymerization, and optical limiting devices that protect sensors from intense laser pulses.

A truly free electron cannot absorb a photon directly (whether one or two photons) without violating conservation of momentum. However, electrons in bound states within atoms, molecules, or semiconductors can participate in TPA. In semiconductors, TPA can cause absorption of sub-bandgap photons, which is a critical consideration in high-power laser applications.