Two-Photon Absorption Calculator

TPA Calculator Overview

The Two-Photon Absorption Calculator (also referred to as a TPA Excitation Rate Calculator or Photon Flux Calculator) lets you determine how many two‑photon excitation events occur per molecule under a given laser configuration. By entering a few experimental parameters, you obtain both the photon flux at the centre of a focused Gaussian beam and the resulting molecular excitation number, making it a straightforward tool for planning two‑photon experiments.

What Is Two‑Photon Absorption?

Two‑photon absorption (TPA) is a nonlinear process in which a molecule simultaneously absorbs two incoming photons. The two photons can share the same energy or have different energies; their total energy must match the gap between the molecule’s initial (usually ground) and final excited state. The absorption proceeds through a short‑lived virtual intermediate state, which does not correspond to a real energy level of the molecule.

First predicted theoretically by Maria Goeppert‑Mayer in 1931, TPA was experimentally confirmed by Kaiser and Garrett in 1963. Today it underpins advanced techniques such as multiphoton microscopy, three‑dimensional data storage, and photodynamic therapy.

TPA Excitation Rate Formula

The number of excitations per molecule $N$ (dimensionless) is calculated with:

$$N = \frac{1}{2}\, \delta\, \tau\, \phi^{2}$$

where:

• $\delta$ — two‑photon absorption cross‑section, expressed in Goeppert‑Mayer (GM) units. $1\ \text{GM} = 10^{-50}\ \text{cm}^{4}\,\text{s}\,\text{ph}^{-1}$.
• $\tau$ — exposure time (s).
• $\phi$ — photon flux at the centre of the Gaussian beam (photons · cm⁻² · s⁻¹).

The factor $\frac{1}{2}$ accounts for the indistinguishability of the two absorbed photons and ensures correct counting of individual excitation events. The cross‑section $\delta$ is a measure of how “effectively” a molecule can undergo TPA; the unit GM honours Maria Goeppert‑Mayer’s pioneering work.

Relating Photon Flux to Laser Parameters

Photon flux is obtained from the beam intensity $I$ and the photon energy $h\nu$:

$$\phi = \frac{I}{h\nu}$$

For a Gaussian laser beam, the peak intensity $I_{0}$ is related to the total power $P$ and the beam radius $w$ (defined at $1/e^{2}$ of the maximum intensity) by:

$$I_{0} = \frac{2P}{\pi w^{2}}$$

The beam radius is often derived from the measured full width at half maximum (FWHM) of the focal spot:

$$w = \frac{\text{FWHM}}{\sqrt{2\ln 2}}$$

Combining these equations, the calculator converts the laser power, wavelength, and beam FWHM into a precise value of $\phi$, which is then used together with $\delta$ and $\tau$ to compute $N$.

Worked Example – Using the TPA Calculator

Suppose a sample is exposed for $1\ \text{s}$ to a laser delivering $10\ \text{W}$ at a wavelength of $840\ \text{nm}$. The molecule’s TPA cross‑section is $210\ \text{GM}$, and the focused beam has an FWHM of $20\ \mu\text{m}$.

Step‑by‑step input:

1. Set $\delta = 210\ \text{GM}$.
2. Enter the laser power $10\ \text{W}$, wavelength $840\ \text{nm}$, and FWHM $20\ \mu\text{m}$.
3. Specify the exposure time $\tau = 1\ \text{s}$.

The tool then evaluates:

$$\phi \approx 9.33 \times 10^{24}\ \text{ph·cm}^{-2}\text{·s}^{-1}$$

and

$$N \approx 91.4$$

These numbers show how efficiently the laser source can drive two‑photon transitions in the chosen molecule.

Applications of Two‑Photon Absorption

TPA is invaluable in several fields:

• High‑resolution bioimaging – Multiphoton microscopy enables deep, three‑dimensional imaging of living tissues with reduced photobleaching and phototoxicity.
• Material characterisation – Measuring TPA cross‑sections helps relate molecular structure to electronic and optical properties.
• Photodynamic therapy – TPA can activate photosensitisers with high spatial precision, improving treatment selectivity.

By bundling the photon flux and excitation rate into a single calculation, this free tool streamlines the design of experiments that rely on two‑photon processes.