# Two-Photon Absorption Calculator

Created by Luciano Mino
Last updated: Jun 26, 2022

Using the two-photon absorption calculator, you can find the amount of two-photon excitations per molecule given a Gaussian beam laser source.

In this short article, we will explain:

• What two-photon absorption is; and
• The two-photon absorption equation.

## What is two-photon absorption?

Two-photon absorption is a phenomenon discovered by Maria Goppert-Mayer in 1931. In this scenario, an atom or molecule absorbs two photons at once, taking the particle from the ground state ($E_{0}$)to a higher virtual energy state ($E_{n}$).

These photons can have equal or different wavelengths, and the difference between the in the two states matches the sum of the energy of both photons.

## Two photon absorption equation

The two-photon absorption calculator finds the number of two-photon excitations per molecule $N$ by using the following formula:

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

where:

• $\delta$ – Cross-section in GM. One GM is $10^{-50}\ \rm cm^4 \cdot s \cdot ph^{-1}$;
• $\tau$ – Exposure time; and
• $\phi$ – Photon flux at the center of the Gaussian beam.

### Finding two-photon excitations number without photon flux

You'll notice the two-photon absorption calculator has a few other parameters.
These come in handy if you don't know $\phi$ since you can calculate it with the following equation:

$\quad \phi = \frac{I}{h\nu} = \frac{I \lambda}{hc}$

where:

• $\nu$ and $\lambda$ are the photon's frequency and wavelength, respectively;
• $I$ is the beam's intensity; and
• $c$ is the speed of light.

And we can write the intensity using its $P$ and beam radius $w$ as:

$\quad I = \frac{2P}{\pi w^2}$

Lastly, we can replace the beam radius with the laser's full width at half-maximum (FWHM) value:

$\quad w = \frac{\rm FWHM}{\sqrt{2\ \ln 2}}$

## Using the two photon absorption calculator

Let's assume we have the following data:

• $\phi = 7.4\cdot 10^{24} \frac{ph}{cm^{2}s}$ for a given laser.
• $\delta = 200\ \text{GM}$.
• $\tau = 1.2 s$

This is all we need to find the number of two-photon excitations according to the two-photon absorption equation.

We can plug that information into the calculator to find that $N = 65.71$.

Luciano Mino
Cross-section (δ)
GM
Laser power (P)
W
Wavelength (λ)
nm
Focus size FWHM
μm
Exposure time (τ)
ns
Photon flux (ϕ)
x10²⁴
ph/(cm²•s)
Excitations per molecule (N)
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