# Two-Photon Absorption Calculator

Created by Purnima Singh, PhD
Reviewed by Steven Wooding
Last updated: Feb 02, 2023

Omni's two-photon absorption calculator allows you to determine the number of two-photon excitations per molecule for a given laser source.

Continue reading this article to know what two-photon absorption (TPA) is and how to calculate the excitation rate using the two-photon absorption equation. You will also find an example of TPA calculation.

## What is two-photon absorption?

Two-photon absorption (TPA) is a process where an atom or a molecule absorbs two photons simultaneously. The energy of the two photons can be identical or different. This absorption results in the excitation of the atom or molecule from one energy state (usually the ground state) to a higher-energy state via an induced virtual state (shown by the dashed horizontal line in figure 1).

As shown in figure 1, the difference in the energy of the two states is equal to the sum of the energy of the two absorbed photons.

$\quad E_n - E_0 = 2\frac{hc}{\lambda}$

The phenomenon was first predicted by Maria Goppert-Mayer in 1931, and Kaiser and Garret experimentally verified it in 1963.

## How to calculate two-photon excitation rate – Two-photon absorption equation

To calculate the number of two-photon excitations per molecule $N$, we will use the formula:

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

where:

• $\delta$ – Two-photon absorption (TPA) cross-section measured 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 a Gaussian beam.

💡 Do you know the two-photon absorption cross-section is expressed in units of GM to honor Maria Goppert-Mayer?

Since the energy carried by a photon is $h\nu$, the number of photons crossing a unit area per unit time (i.e., the photon flux) is related to the intensity of the beam $I$ by the expression:

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

The intensity of the laser beam with power $P$ and the beam radius $w$ can be described as:

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

where the beam radius is related to the full width at half maximum (FWHM) as (see figure 2):

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

In the next section, we will use the two-photon absorption calculator to calculate the number of excitations per molecule for a given laser pulse.

## How to use the two-photon absorption calculator – TPA calculation example

Let's calculate the photon flux and number of excitations per molecule when a sample is irradiated for $1\ \rm s$ with a $10\ \rm W$ laser source of wavelength $840\ \rm nm$. The two-photon absorption cross-section is $210\ \rm GM$, and the FWHM of the focussed laser beam is $20 \ \rm\mu m$.

1. Enter the two-photon absorption cross-section in GM ($210\ \rm GM$).

2. Type the power of the laser source ($10\ \rm W$), the wavelength of the beam ($840\ \rm nm$), and the FWHM of the focussed beam ($20 \ \mu \rm m$). You can also use the wavelength calculator or the energy to wavelength calculator to find out the wavelength if you know the frequency or energy of the laser source.

3. Enter the exposure time ($1\ \rm s$).

4. The two-photon absorption calculator will give you the photon flux at the center of the beam ($9.33 \times 10^{24}\ \rm ph/(cm^2.s)$) and the number of excitations per molecule ($91.4$).

## FAQ

### What does photon absorption mean?

Photon absorption is a process in which an atomic electron absorbs the energy of an incident photon. If the photon's energy is higher than the binding energy of the electron, the electron is ejected from the atom. Otherwise, the electron gets excited to a higher energy state within the atom.

### Can a free electron absorb a photon?

No, a free electron cannot absorb or emit a photon. The conditions for conservation of energy and momentum are not satisfied in the process if the electron is free. Hence, only electrons that are bound to atoms can absorb photons.

### How do you measure two-photon absorption cross-section?

Some of the techniques used to measure two-photon absorption cross-section are:

• Two-photon excited fluorescence (TPEF) spectroscopy;
• Z-scan approach;
• Mass-sedimentation approach; and
• Nonlinear transmission method.

### What are the applications of two-photon absorption?

The two-photon absorption technique has several applications, including:

• The study of novel materials and investigate the relationship between their molecular structure and electronic and optical properties.

• Performing high-resolution imaging of live cell and tissue samples.

Purnima Singh, PhD
Cross-section (δ)
GM
Laser power (P)
W
Wavelength (λ)
nm
Focus size FWHM
μm
Exposure time (τ)
ns
Photon flux (ϕ)
×10²⁴
ph/(cm²•s)
Excitations per molecule (N)
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