In ordinary (one-photon) absorption a molecule takes up a single photon whose energy matches the gap between two electronic states. In two-photon absorption (TPA) it takes up two photons at the same time, each carrying about half the energy – two 800 nm near-infrared photons can do the job of one 400 nm violet photon. Maria Göppert-Mayer predicted the effect in her 1931 doctoral thesis; it was only observed after the laser was invented.
Because two photons must arrive together, the excitation rate grows with the square of the light intensity. That is the key to two-photon microscopy and 3D laser micro-printing: excitation happens only in the tiny, intense focal spot, so a laser can image or polymerise a point deep inside a sample without affecting the material above and below it. Near-infrared light also scatters less in tissue.
The strength of the effect is the two-photon cross-section δ, measured in Göppert-Mayer units (1 GM = 10⁻⁵⁰ cm⁴·s per photon). Common fluorescent dyes have δ of roughly 1–300 GM; specially designed chromophores reach thousands. Femtosecond pulsed lasers are used because concentrating the energy in short pulses multiplies the average excitation rate by the inverse of the duty cycle.
How to use the Two-Photon Absorption Calculator
- Two-photon absorption cross-section δ
- Laser
- Laser power (average power for a pulsed laser)
- Wavelength λ
- Focal spot size (FWHM)
- Exposure time
- Pulse repetition rate
- Pulse duration
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
The beam is treated as a Gaussian spot. The FWHM is converted to the 1/e² radius w, the peak intensity at the centre is 2P/(πw²), and dividing by the photon energy hc/λ gives the photon flux φ. Each molecule is excited at the rate ½·δ·φ²; multiplying by the illumination time gives the excitations per molecule. For pulsed lasers the peak power during each (assumed rectangular) pulse is used and only the time the pulses are on counts.
Worked example
Continuous-wave laser: δ = 210 GM, P = 10 W, λ = 840 nm, FWHM = 20 µm, 1 s exposure. The 1/e² radius is w = 20 / 1.1774 = 16.99 µm = 1.699 × 10⁻³ cm. Peak intensity I = 2 × 10 / (π × (1.699 × 10⁻³)²) = 2.206 × 10⁶ W/cm². One 840 nm photon carries hc/λ = 2.365 × 10⁻¹⁹ J (1.476 eV), so φ = 9.33 × 10²⁴ photons/(cm²·s). Each molecule is excited ½ × 210 × 10⁻⁵⁰ × (9.33 × 10²⁴)² = 91.4 times per second, i.e. about 91 excitations in 1 s.
Pulsed laser: δ = 10 GM, 10 mW average at 80 MHz with 100 fs pulses, λ = 800 nm, FWHM = 0.5 µm, 1 ms exposure. The peak power is 0.010 / (80 × 10⁶ × 100 × 10⁻¹⁵) = 1,250 W, w = 0.4247 µm, I = 4.41 × 10¹¹ W/cm² and φ = 1.78 × 10³⁰ photons/(cm²·s). Each pulse excites a molecule with probability 0.0158 – below saturation – and 1 ms (80,000 pulses) gives about 1,263 excitations per molecule.
When to use it
Use the result to compare dyes and laser settings, to check whether excitation is far from saturation (fewer than about 0.1 excitations per pulse), and to estimate how long a 3D-printing voxel or an imaging pixel needs to be exposed. The model assumes a Gaussian focus, rectangular pulses and no ground-state depletion; real Gaussian-shaped pulses give a temporal correction factor of roughly 0.6–0.7, and the axial shape of the focus is ignored.
Tips
- Keep temperature in kelvin and pressure in consistent units.
- For gases, ideal-gas behaviour is a good approximation near room conditions.
- Half-life and rate constant are linked by t½ = ln 2 / k for first-order processes.
- Everything is calculated in your browser.
Common mistakes to avoid
- Using the mass number instead of the molar mass in Graham’s law.
- Mixing partial pressures and mole fractions.
- Forgetting that osmotic pressure uses the total particle concentration (van ‘t Hoff factor).
Inputs explained
| Field | What to enter |
|---|---|
| Two-photon absorption cross-section δ | Units: GM, cm⁴·s/photon |
| Laser | Choose one: Continuous wave (CW), Pulsed (e.g. femtosecond Ti:sapphire) |
| Laser power (average power for a pulsed laser) | Units: W, mW, kW, µW |
| Wavelength λ | Units: nm, µm, Å |
| Focal spot size (FWHM) | Units: µm, nm, mm |
| Exposure time | Units: s, ms, µs, min, h |
| Pulse repetition rate | Units: MHz, kHz, Hz, GHz |
| Pulse duration | Units: fs, ps, ns |
What each result means
- Beam radius at 1/e² (w) = [Focal spot size (FWHM)]÷sqrt(2×ln(2))×1e6
- Peak intensity at the beam centre = 2×[Peak power]÷(pi×([Beam radius at 1÷e² (w)]×1e-4)^2)
- Photon energy = 6.62607015e-34×299792458÷[Wavelength λ]÷1.602176634e-19
- Photon flux at the centre φ = [Peak intensity at the beam centre]÷(6.62607015e-34×299792458÷[Wavelength λ])
- Excitation rate per molecule (while the light is on) = 0.5×[Two-photon absorption cross-section δ]×1e-50×[Photon flux at the centre φ]^2
- Excitations per molecule per pulse = [Excitation rate per molecule (while the light is on)]×[Pulse duration]
- Equivalent one-photon wavelength (λ/2) = [Wavelength λ]÷2×1e9
Example values
How the main result changes when Two-photon absorption cross-section δ changes and everything else stays at the example values above:
| Two-photon absorption cross-section δ (GM) | Excitations per molecule during the exposure |
|---|---|
| 105 | 45.6999 |
| 157.5 | 68.5499 |
| 210 | 91.3999 |
| 262.5 | 114.2498 |
| 315 | 137.0998 |
| 420 | 182.7997 |
Formula
w = FWHM ÷ √(2 ln 2) · I = 2P ÷ (π w²) · φ = I·λ ÷ (h·c) · rate = ½ · δ · φ² · N = rate × exposure time (pulsed: P_peak = P_avg ÷ (f·τ_p), N = rate × τ_p × f × t)Worked example
Inputs — Two-photon absorption cross-section δ: 210 GM; Laser: Continuous wave (CW); Laser power (average power for a pulsed laser): 10 W; Wavelength λ: 840 nm; Focal spot size (FWHM): 20 µm; Exposure time: 1 s.
Results — Beam radius at 1/e² (w) = 16.9864 µm; Peak power = 10 W; Peak intensity at the beam centre = 2,206,356.0015 W/cm²; Photon energy = 1.476 eV; Photon flux at the centre φ = 9.3299e24 photons/(cm²·s); Excitation rate per molecule (while the light is on) = 91.3999 per s; Excitations per molecule during the exposure = 91.3999; Equivalent one-photon wavelength (λ/2) = 420 nm.
Key terms
- Two-photon absorption
- Simultaneous absorption of two photons whose energies add up to the transition energy.
- Cross-section δ
- Molecular constant giving the two-photon absorption probability, in GM.
- Photon flux φ
- Number of photons crossing unit area per unit time, photons/(cm²·s).
- FWHM
- Full width at half maximum – the diameter of the spot where intensity is half its peak.
- Duty cycle
- Fraction of time a pulsed laser is emitting: repetition rate × pulse duration.
Frequently asked questions
What is a GM unit?
The Göppert-Mayer unit of two-photon cross-section: 1 GM = 10⁻⁵⁰ cm⁴·s·photon⁻¹. Fluorescein has about 38 GM near 780 nm and rhodamine B about 200 GM near 840 nm in published measurements.
Why does two-photon absorption depend on intensity squared?
Two photons must be absorbed at the same moment, so the probability is proportional to the product of the two photon fluxes, φ². Doubling the laser power quadruples the excitation rate.
Why are femtosecond lasers used?
For the same average power, squeezing the light into 100 fs pulses at 80 MHz raises the peak power by 1/(f·τ) = 125,000 times. Because the rate goes as power squared, the time-averaged excitation rises by the same factor while the sample heating (set by average power) stays the same.
How is the focal spot size related to the numerical aperture?
For a diffraction-limited focus the lateral FWHM is roughly 0.51·λ/NA (one-photon) – about 0.3–0.5 µm for an oil-immersion objective at 800 nm. Smaller spots give much higher intensity.
How many excitations per pulse is too many?
Above roughly 0.1 excitations per molecule per pulse the ground state starts to deplete (saturation) and photobleaching rises, so the ½δφ² formula overestimates the signal.
What does the beam radius w mean?
It is the 1/e² intensity radius of a Gaussian beam; FWHM = w·√(2 ln 2) ≈ 1.177·w.
What is two-photon absorption used for?
Two-photon fluorescence microscopy, 3D microfabrication (two-photon polymerisation), optical limiting, photodynamic therapy and 3D optical data storage.