Last updated: July 3, 2026
Two-Photon Absorption Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Quick Answer
The two-photon absorption calculator converts σGM in Göppert-Mayer units to σ = σGM × 10⁻⁵⁰ cm⁴·s·photon⁻¹, computes the equivalent one-photon wavelength λ/2, and calculates the combined two-photon energy 2hc/λ. For 100 GM at 800 nm, σ = 1 × 10⁻⁴⁸ cm⁴·s/photon, λeq = 400 nm, and E₂γ ≈ 4.966 × 10⁻¹⁹ J.
To convert a two-photon absorption cross-section from GM units, multiply by ten to the minus fifty. The equivalent one-photon wavelength is half the excitation wavelength, and the combined two-photon energy is two h c divided by lambda.
Key Takeaways
- One Göppert-Mayer unit equals 10⁻⁵⁰ cm⁴·s·photon⁻¹.
- A 100 GM cross-section converts to 1 × 10⁻⁴⁸ cm⁴·s/photon.
- Two photons at wavelength λ combine to the energy of one photon at λ/2.
- At 800 nm, the equivalent single-photon wavelength is 400 nm and E₂γ ≈ 4.966 × 10⁻¹⁹ J.
- Bulk TPA coefficients require concentration or number density in addition to molecular cross-section.
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
σ = σGM × 10⁻⁵⁰; λeq = λ/2; E2γ = 2hc/λ; b = σ/E2γ
Where:
- σGM=Two-photon absorption cross-section(GM)
- σ=Two-photon cross-section in cgs-SI convention(cm⁴·s·photon⁻¹)
- λ=Excitation wavelength(nm)
- λeq=Equivalent single-photon wavelength(nm)
- E2γ=Combined energy of the two absorbed photons(J)
- h=Planck constant(J·s)
- c=Speed of light(m/s)
Worked Examples
100 GM fluorophore at 800 nm
A common benchmark for nonlinear fluorophores excited with a Ti:sapphire laser.
- 1Convert GM to cross-section: σ = 100 × 10⁻⁵⁰ = 1 × 10⁻⁴⁸ cm⁴·s/photon.
- 2Equivalent one-photon wavelength: λeq = 800 nm ÷ 2 = 400 nm.
- 3Combined photon-pair energy: E2γ = 2hc/λ = 4.966 × 10⁻¹⁹ J.
1 GM reference at 800 nm
The defining Göppert-Mayer unit converted directly into cm⁴·s/photon.
- 1Apply the unit definition: 1 GM = 1 × 10⁻⁵⁰ cm⁴·s/photon.
- 2Use λeq = λ/2, so 800 nm becomes 400 nm.
- 3The photon-pair energy is the same as any 800 nm two-photon excitation: 4.966 × 10⁻¹⁹ J.
High cross-section probe at 920 nm
A bright two-photon probe with a larger cross-section and longer excitation wavelength.
- 1Convert cross-section: σ = 250 × 10⁻⁵⁰ = 2.5 × 10⁻⁴⁸ cm⁴·s/photon.
- 2Equivalent one-photon wavelength: λeq = 920 nm ÷ 2 = 460 nm.
- 3Longer wavelength lowers the combined two-photon energy to about 4.32 × 10⁻¹⁹ J.
Introduction
Two-photon absorption (TPA) is a nonlinear optical process in which a molecule absorbs two lower-energy photons nearly simultaneously to reach an excited state normally associated with one higher-energy photon. This calculator converts a TPA cross-section from Göppert-Mayer units into cm⁴·s·photon⁻¹, finds the equivalent one-photon wavelength, and estimates a wavelength-dependent per-molecule coefficient from the two-photon energy. It complements spectroscopy tools such as the Beer-Lambert law calculator and concentration workflows using the molarity calculator.
What is two-photon absorption?
In ordinary absorption, one photon supplies the transition energy. In two-photon absorption, two photons combine their energies during a very short virtual-state interaction. Because the probability depends on the square of light intensity, TPA is most visible with focused ultrafast lasers and is central to two-photon microscopy, microfabrication, and nonlinear spectroscopy.
Two photons of wavelength λ act like one photon of wavelength λ/2 for energy matching.
The event is nonlinear: doubling intensity can roughly quadruple the absorption rate in the unsaturated regime.
TPA cross-sections are usually tabulated in GM units because the cgs-scale numbers are extremely small.
GM unit conversion
The Göppert-Mayer unit is defined for two-photon cross-sections as 1 GM = 10⁻⁵⁰ cm⁴·s·photon⁻¹. The calculator therefore multiplies the entered σGM value by 10⁻⁵⁰. For example, 100 GM becomes 1 × 10⁻⁴⁸ cm⁴·s/photon. This unit is commonly used in nonlinear optics papers because molecule-scale TPA probabilities are far smaller than one-photon extinction coefficients.
Do not compare a GM value directly with a Beer–Lambert molar absorptivity; the units and intensity dependence are different.
Two-photon energy and equivalent wavelength
For excitation wavelength λ, the combined absorbed energy is E₂γ = 2hc/λ. Using the exact SI constants h = 6.62607015 × 10⁻³⁴ J·s and c = 2.99792458 × 10⁸ m/s, 800 nm excitation gives E₂γ = 4.966 × 10⁻¹⁹ J. The equivalent single-photon wavelength is λ/2, so 800 nm two-photon excitation corresponds energetically to a 400 nm one-photon transition. See the NIST constants database for the exact constant values.
Always convert nanometres to metres before using h and c in SI units.
Interpreting the TPA coefficient
This calculator reports a per-molecule coefficient b = σ/E₂γ in cm⁴/J. It is useful as a wavelength-normalized intensity coefficient, but bulk two-photon absorption coefficients used in materials optics also require number density, concentration, pulse shape, and convention choices. For solution experiments, use the concentration calculator or calibration curve calculator to manage sample concentration separately.
Cross-section σ describes an individual absorber.
Bulk β values require absorber density or molar concentration.
Pulsed laser measurements can depend on temporal and spatial beam profiles.
Typical cross-section ranges
Small organic molecules may have TPA cross-sections below 1 GM, fluorescent dyes often fall from tens to hundreds of GM, and specially designed chromophores or nanoparticles can be much larger. The exact value depends strongly on wavelength, solvent, polarization, and measurement method.
| System | Typical σGM | Notes |
|---|---|---|
| Weak organic chromophore | < 1–10 GM | Often difficult to measure precisely |
| Common fluorescent dye | 10–300 GM | Used in two-photon microscopy |
| Optimized donor-acceptor dye | 100–1000+ GM | Strong wavelength dependence |
| Nanomaterial probe | reported very large | Check normalization and particle definition |
Assumptions and limitations
The calculation is a unit and energy conversion; it does not model saturation, photobleaching, pulse repetition rate, focal volume, quantum yield, or sample scattering. TPA data should be interpreted with the experimental convention used by the source. Reviews such as Makarov et al. 2008 and Xu and Webb 1996 discuss measurement methods and calibration standards.
Use a cross-section measured at or near your excitation wavelength.
Report solvent, polarization, pulse duration, and reference standard when comparing values.
Avoid extrapolating TPA spectra far outside measured wavelengths.
Very high intensities may introduce excited-state absorption or damage.
Quick Reference Card
Two-Photon Absorption — Quick Reference
Quick reference • Two-Photon Absorption Calculator
σ = σGM × 10⁻⁵⁰; λeq = λ/2; E₂γ = 2hc/λValid range: Use positive σGM and wavelength; common molecular data span <1 to >1000 GM depending on chromophore and wavelength.
Common Values
⚠ Watch Out
- •Use wavelength-specific σGM data; TPA spectra can vary sharply with wavelength.
- •Do not compare GM values directly with one-photon molar absorptivity.
- •Bulk β values need concentration or number density, not only molecular σ.
- •Check whether literature values are per molecule, per particle, or per chromophore.
- •High laser intensity can cause saturation, bleaching, or other nonlinear processes.
Pro Tips
- →Quote both σGM and the excitation wavelength when reporting TPA data.
- →Use relative tolerances when testing tiny cross-sections such as 10⁻⁵⁰.
- →Convert wavelength from nm to m before computing energy with SI constants.
- →Pair TPA calculations with concentration and calibration tools for solution experiments.
- →Compare measurements only when pulse width, reference standard, and solvent are compatible.
FAQs
What does GM mean in two-photon absorption?
GM stands for Göppert-Mayer, the conventional unit for a two-photon absorption cross-section. One GM equals 10⁻⁵⁰ cm⁴·s·photon⁻¹.
How do I convert 100 GM to cm⁴·s/photon?
Multiply by 10⁻⁵⁰. Thus 100 GM = 100 × 10⁻⁵⁰ = 1 × 10⁻⁴⁸ cm⁴·s/photon.
Why is the equivalent wavelength half of the excitation wavelength?
Two photons each contribute energy hc/λ, so together they provide 2hc/λ. That equals the energy of one photon with wavelength λ/2.
What is the two-photon energy at 800 nm?
Using E₂γ = 2hc/λ, 800 nm excitation gives approximately 4.966 × 10⁻¹⁹ J for the combined photon pair.
Is cross-sectionSI a true SI unit?
The conventional converted unit used here is cm⁴·s·photon⁻¹, following the GM definition. It is not expressed in pure metre-based SI, but it is the standard reporting form in TPA literature.
Can this calculator give a bulk β coefficient for a solution?
It gives a per-molecule coefficient from σ and photon-pair energy. A bulk material β also needs absorber concentration or number density and the experimental convention.