Two-Photon Absorption Calculator

Estimate two-photon excitations per molecule from laser and focus inputs.

Seven unit-aware inputs, with backward solves for power, wavelength, focus size, flux, cross-section and exposure time.

Updated October 17, 2026

CreatorFrank Zhao

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What the calculator answers

Two-photon absorption only happens when two photons reach a molecule within the same instant, so the rate follows the square of the photon flux rather than the number of photons that happen to pass by. This page turns your laser and focus settings into the photon flux at the centre of the focus and the average number of excitations each molecule collects while the beam is on it.

Those two numbers answer the questions that come up around a two-photon experiment: is this dye bright enough at the wavelength I have, how much power puts the sample into a useful regime, and does a longer dwell push the answer past the point where the model still means anything? Seven quantities are linked here — cross-section, power, wavelength, focus size, exposure time, photon flux and excitations per molecule — and the page will solve for whichever one you leave out.

The same figures are used well beyond imaging. Two-photon cross-sections are compared in Goeppert-Mayer units throughout the photophysics literature, damage and safety discussions revolve around intensity and photon flux, and a course exercise often needs a worked excitation count next to a measured fluorescence signal.

7 linked quantities

Fill six and the seventh is solved

Photon flux

At the centre of the focus, ph/(cm²·s)

Excitations per molecule

Averaged over the exposure you enter

Quick start

Enter the sample and the beam

The cross-section is the only value you normally have to look up: it describes how willing your dye or material is to absorb two photons at the wavelength you are using, and it is quoted in Goeppert-Mayer units. The other three come from your equipment — the power your meter reads, the laser wavelength, and the full width at half maximum of the focused spot in micrometres.

Say how long the sample is exposed

Enter the interval you actually care about: one pixel dwell time, one frame, or a full second. The field accepts picoseconds through weeks, and only the duration matters — a 10 µs dwell behaves exactly like 0.00001 s.

Read the photon flux, then the excitations

The flux is reported on the ×10²⁴ scale, so a value of 9.32992 means 9.32992 × 10²⁴ photons crossing one square centimetre of the focal plane every second. The excitations per molecule is the headline number: how many two-photon events each molecule averages over the interval you entered.

Check the working on a setup you can picture

A dye with a 210 GM cross-section, a 10 W beam at 840 nm, focused to a 20 µm FWHM spot, left on the sample for one second:

\phi

=

4.4428\times10^{20}\cdot\frac{10\times8.4\times10^{-7}}{(2\times10^{-5})^{2}}

=

9.32992\times10^{24}\ \mathrm{ph/(cm^{2}{\cdot}s)}

N

=

\tfrac{1}{2}\times210\times(9.32992\times10^{24})^{2}\times1\times10^{-50}

=

91.4

Ninety-one excitations per molecule in a single second is a hard drive, not a gentle measurement: the molecule is being cycled through its excited state many times, which is exactly where saturation and photobleaching start to shape the signal. For imaging work the useful range is usually well under one excitation per molecule per pixel, which is what the worked examples below show.

The two formulas

Everything on the page comes from two short equations. The first gives the photon flux at the centre of a focused Gaussian beam:

\phi = 10^{-4}\cdot\frac{4\ln 2}{\pi h c}\cdot\frac{P\,\lambda}{\mathrm{FWHM}^{2}}

The second converts that flux into an excitation count per molecule:

N = \frac{1}{2}\,\delta\,\phi^{2}\,\tau \times 10^{-50}

Notice the square on the flux. Two photons have to meet, so the rate grows with the product of the photon density with itself: double the flux and each molecule absorbs four times as often. That squared dependence is also why the focus size matters so much — halving the FWHM quadruples the flux and multiplies the excitation count by sixteen.

Where the flux coefficient comes from

The coefficient in the first equation is not an arbitrary fit. A Gaussian spot reported by its full width at half maximum has a beam radius of

w = \mathrm{FWHM}\big/\sqrt{2\ln 2}

\approx \mathrm{FWHM}/1.1774

the intensity at the centre is $I = 2P/(\pi w^{2})$ , and the flux of photons in a beam of that intensity is $\phi = I\lambda/(hc)$ . Substituting the first two into the third collapses the whole chain into $4\ln 2/(\pi hc)$ , and the $10^{-4}$ converts square metres into square centimetres. Written in watts, metres and seconds, the coefficient is $4.4428\times10^{20}$ .

The constants are exact, and so is the 4 ln 2 factor

The Planck constant is defined as 6.626 070 15 × 10⁻³⁴ J Hz⁻¹ and the speed of light as 299 792 458 m s⁻¹, both with zero uncertainty, so this page uses those exact values rather than a rounded table entry.

The coefficient is also built from the defining factor $4\ln 2$ rather than from a rounded $1.17741$ . The difference in the final flux is about 4 parts in 10⁸ — invisible next to any real measurement, but it means the expression above is the exact one rather than an approximation of itself.

What the symbols mean

\phi

Photon flux at the centre of the focus, in photons per square centimetre per second

N

Average number of two-photon excitations per molecule over the exposure

\delta

Two-photon absorption cross-section, in Goeppert-Mayer units

P

Laser power reaching the sample, in watts

\lambda

Laser wavelength in vacuum

\mathrm{FWHM}

Full width at half maximum of the focused spot

\tau

Exposure time — the total interval the sample sits in the beam

h,\ c

Planck constant and speed of light in vacuum, both exact

\mathrm{GM}

1 GM = 10⁻⁵⁰ cm⁴·s per photon, the unit named for Maria Goeppert-Mayer

Units are normalised before the formulas run

Power is converted to watts, wavelength and focus size to metres and exposure time to seconds, so any of the supported scales — milliwatts, nanometres, micrometres, picoseconds and the rest — give the same underlying value. A cross-section in GM is multiplied by 10⁻⁵⁰ on the way in, which is exactly what the trailing factor in the second equation does. The ×10ⁿ setting on the flux is a display scale: the flux itself is always in photons per square centimetre per second.

What can be solved backwards

Both equations rearrange for each of their four quantities, and the two share the photon flux. That is what lets a target excitation count drive the rest of the page: give the cross-section, the dwell time and the focus, and the flux is recovered from the second equation and handed to the first one, which then returns the power required. The same chain runs the other way — from laser settings to a predicted excitation count.

One related quantity is not on the page: the energy of the two-photon transition itself, $2hc/\lambda$ . At 920 nm that is 4.3184 × 10⁻¹⁹ J, or 2.6953 eV, the same energy as a single photon near 460 nm. If you need it in electronvolts, wavenumbers or kelvin, the Energy Conversion Calculator takes that joule value and returns any of them.

Worked examples

A microscopy dwell at 920 nm

Situation: A mode-locked source delivers 1.5 W at 920 nm to a dye with a 150 GM cross-section, focused to a 1.2 µm FWHM spot. You want to know what one 10 µs pixel dwell does to the sample.

Entered

Cross-section150 GM

Laser power1.5 W

Wavelength920 nm

Focus FWHM1.2 µm

Exposure time10 µs

What the page returns

\phi = 4.2577\times10^{26}\ \mathrm{ph/(cm^{2}{\cdot}s)}

\qquad N = 1.3596

The flux field reads 425.77 on its ×10²⁴ scale. Almost all of the change from the previous example comes from the focus: a 1.2 µm spot concentrates the same kind of beam into an area hundreds of times smaller.

In practice: That is roughly one excitation per molecule per pixel — the top of the linear range, where the signal is strong but the excited-state population is still small.

Working backwards to a power setting

Situation: Same dye, same 920 nm source, same 1.2 µm focus and 10 µs dwell, but this time you fix the answer first: you want one excitation per molecule per pixel and need to know what average power that takes.

Entered

Excitations per molecule1

Cross-section150 GM

Wavelength920 nm

Focus FWHM1.2 µm

Exposure time10 µs

What the page returns

\phi = 3.65148\times10^{26}\ \mathrm{ph/(cm^{2}{\cdot}s)}

\qquad P = 1.28643\ \mathrm{W}

This is the bidirectional part of the page at work. The excitation equation is solved for the flux first, and that flux becomes one of the inputs to the flux equation, which returns the power. Nothing else has to be rearranged by hand.

In practice: Use it as a starting point on the bench, then trim the power until the measured count rate is linear in power — that is the practical sign you are below saturation.

Checking whether the model still applies

Situation: The same 1.5 W, 920 nm, 1.2 µm setup as the first example, but with a ten-second exposure instead of a 10 µs dwell.

Entered

Cross-section150 GM

Laser power1.5 W

Wavelength920 nm

Focus FWHM1.2 µm

Exposure time10 s

What the page returns

N = 1.36\times10^{6}\ \text{(flagged)}

The number is still shown, but the field turns red with an explanation. A million excitations per molecule means the ground state is largely used up, and the formula this page evaluates assumes it is not.

In practice: When the count reaches this level, treat the result as an upper bound: reduce the power or the exposure, or measure the fluorescence rather than trusting the arithmetic.

Inputs you can trust

Measure the power where the sample sits

Objectives, dichroic mirrors and scan optics all remove part of the beam, so the number printed on the laser head is not the power at the focus. Put a power meter in the sample plane, or at least after the last optic, and enter that figure.

Convert your beam radius to a FWHM

Beam profilers often report the 1/e² radius $w$ rather than the full width at half maximum. For a Gaussian beam the two differ by a fixed factor: $\mathrm{FWHM} = 1.1774\,w$ . Entering the radius unchanged understates the spot and overstates the flux by about 39%.

Decide which interval you are describing

The exposure time is the interval the calculation averages over, and the answer scales directly with it. A single pulse, a pixel dwell and a full second are three different questions. For a pulsed source you can also describe one pulse on its own: the peak power is roughly the average power divided by the repetition rate and the pulse width.

Use a cross-section measured at your wavelength

Two-photon cross-sections vary strongly with wavelength, and a value quoted at 800 nm can be several times off at 920 nm. Take the number from a measurement made at, or close to, the wavelength you are actually using, and treat it as the largest uncertainty in the answer.

Keep one eye on the magnitude

A flux in the 10²⁵ to 10²⁶ range is routine for a focused femtosecond beam; an excitation count of a few per molecule is already a lot. If the count per dwell comes out in the thousands, the sample is being driven far past the linear regime and the measured brightness will not follow the prediction.

Cross-check with what you see

Fluorescence is proportional to the excitation count while the sample stays below saturation, so a quick power series on the bench is the simplest test of any prediction here: the signal should double when the power doubles. When it stops doing that, the model below is no longer the right description.

One worked pulse example makes the interval question concrete. A source delivering 1.5 W average power at an 80 MHz repetition rate with 100 fs pulses has a peak power near 1.875 × 10⁵ W. Entering that peak power and a 0.1 ps pulse with the 150 GM, 920 nm, 1.2 µm figures above gives a peak flux of 5.32 × 10³¹ ph/(cm²·s) and about 212 excitations per molecule in a single pulse — a reminder that pulse-resolved and time-averaged questions have very different answers.

Questions that come up

Why is the photon flux so large?

Because it counts photons crossing a square centimetre, and a square centimetre is a large area next to a focused spot. A one-watt beam at 840 nm carries around 4 × 10¹⁸ photons per second, and squeezing that into a spot a couple of dozen micrometres across produces a density in the 10²⁴ range. Values like 9.32992 × 10²⁴ ph/(cm²·s) are ordinary for focused laser light, not a sign that something has gone wrong.

Is photon flux the same as intensity?

They measure different things. Intensity is power per unit area, usually quoted in W/cm², and it says how much energy arrives. Photon flux counts the photons themselves, in photons per square centimetre per second, and it is what a photochemical or two-photon rate depends on. The two are linked through the photon energy by $\phi = I\lambda/(hc)$ , so at a fixed wavelength they are proportional to each other.

Which cross-section value should I enter?

A two-photon cross-section measured for the same molecule at the same wavelength, in GM. It is emphatically not the ordinary one-photon cross-section: the two describe different processes and differ by many orders of magnitude. If you only have a value from another wavelength, record it as a rough estimate and expect the answer to shift.

Why does the focus size change the result so much?

The flux goes as $1/\mathrm{FWHM}^{2}$ , and the excitation count goes as the square of the flux. Halving the spot therefore multiplies the flux by four and the excitation count by sixteen. Focus size is usually the single most sensitive input on the page after the cross-section.

Can I use this for a continuous-wave laser?

Yes. Neither formula assumes pulses — a continuous beam simply has a constant flux, and you enter the exposure interval you care about. What changes is the practical outcome: without the high peak intensity of a pulsed source, two-photon rates from a continuous beam are small unless the light is focused very tightly.

Where the model stops

•The beam is assumed to be an ideal Gaussian. Aberrations, a clipped aperture or a non-Gaussian profile make the real peak intensity differ from the value computed from the FWHM.
•The ground state is assumed to be undepleted. Once a molecule has absorbed two photons it is no longer available, so the formula overstates the count as the number grows. The page flags results above about a million excitations per molecule, but saturation starts to matter well before that.
•Power and exposure describe one interval. Mode-locked lasers deliver their light in short, intense pulses, and the peak flux during a pulse is far higher than the average flux over the same period. Enter the peak power and pulse width if you want a pulse-resolved count; entering average power over a long exposure answers a time-averaged question instead.
•The cross-section is treated as a constant. It is assumed to be known at the laser wavelength and to stay unchanged during the exposure. It is also the input users are most likely to take from a different wavelength, which is a common source of large errors.
•No diffraction check is performed. A lens cannot produce a spot much smaller than about half the wavelength, so if your FWHM is below roughly $\lambda/2$ the arithmetic is describing a spot that cannot exist. Nothing on the page will stop you — check the figure against your microscope objective.
•Photobleaching, losses and detection are excluded. The page predicts excitations, not the fluorescence you will collect; quantum yield, collection efficiency and bleaching all sit between the two. Treat the result as a planning figure and confirm it with a power series on the bench.

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