How this instrument works
Two-photon absorption is a nonlinear optical process where a molecule absorbs two photons nearly simultaneously, together supplying the energy that one higher-energy photon would normally supply alone. Because it needs two photons to arrive within an extremely short window, the probability of it happening scales with the square of light intensity rather than linearly — which is exactly why it only becomes significant in the intensely concentrated light at the focal point of a tightly focused, pulsed laser beam, and stays negligible everywhere else in the beam path. This intensity-squared dependence is what gives two-photon microscopy its defining advantage: excitation, and the resulting fluorescence, is confined almost entirely to a tiny focal volume, with none of the out-of-focus background a standard single-photon microscope produces.
A molecule's two-photon absorption efficiency is described by its cross-section, delta, conventionally reported in Goeppert-Mayer units (1 GM = 10^-50 cm^4.s per photon) — common fluorescent dyes and proteins used in microscopy typically fall somewhere in the 1-1000 GM range. Getting from a laser's practical setup parameters to an excitation probability takes a short chain of physics: focal spot size and laser power set the peak intensity at the focus, intensity and wavelength set the photon flux (photons per area per second), and photon flux squared, combined with the cross-section and pulse duration, sets the actual probability that a given molecule absorbs two photons during one pulse.
This is a research-instrument-scale calculation, built for the audience actually setting up or troubleshooting two-photon excitation microscopy or spectroscopy — estimating whether a given laser power, focus, and pulse duration combination will produce a usable excitation rate for a fluorophore of known cross-section. The excitation probability per pulse this calculator reports is typically an extremely small number (often around 10^-9 to 10^-12), which is expected and not an error: real two-photon fluorescence signal builds up from millions of pulses per second at a laser's typical repetition rate, not from any single pulse.
- Enter the fluorophore's two-photon absorption cross-section, in Goeppert-Mayer units, into Two-photon absorption cross-section, delta (GM = 1e-50 cm^4.s/photon).
- Enter the laser's average power into Laser power, P (W).
- Enter the excitation wavelength into Wavelength, lambda (nm) — typically 700-1100 nm for common two-photon microscopy fluorophores.
- Enter the focal spot size (full width at half maximum) into Focal spot size, FWHM (um), and the pulse duration into Exposure time, tau (fs).
- Read Photon flux, phi (photons / (cm^2 x s)) and Excitations per molecule, N below the inputs — N is the estimated probability that a given molecule is excited during one pulse.
Worked example — a common fluorescein-range dye at 800 nm
Enter 100 into Two-photon absorption cross-section, delta (GM) — a typical order-of-magnitude cross-section for common two-photon dyes like fluorescein — 0.01 into Laser power, P (W) (10 mW average power), 800 into Wavelength, lambda (nm) (a standard Ti:Sapphire two-photon wavelength), 0.5 into Focal spot size, FWHM (um) (a diffraction-limited focus), and 100 into Exposure time, tau (fs) (a typical femtosecond pulse width). Photon flux, phi (photons / (cm^2 x s)) reads about 1.42 x 10^25, and Excitations per molecule, N reads about 1.01 x 10^-11.
That N value means roughly a one-in-a-hundred-billion chance that any single molecule at the focal point absorbs two photons during one femtosecond pulse — vanishingly small per pulse, but a Ti:Sapphire laser fires roughly 80 million pulses every second, so across one second of continuous illumination a huge number of independent excitation events still accumulate, which is what produces measurable, usable fluorescence signal in a real two-photon microscopy image.
Questions
Why is the excitation probability, N, such a tiny number?
Because two-photon absorption genuinely is a low-probability event per pulse — it depends on two photons arriving at the same molecule within an extremely short window, which is inherently rare even inside a tightly focused, high-intensity laser pulse. A tiny per-pulse N is normal and expected; real two-photon imaging works because a laser fires many millions of pulses per second, so the total number of excitation events across even one second of imaging is still large despite the vanishingly small probability from any individual pulse.
What is a Goeppert-Mayer (GM) unit, and why is it so small?
It's the conventional unit for reporting a molecule's two-photon absorption cross-section, defined as 10^-50 cm^4.s per photon and named after Maria Goeppert-Mayer, who first predicted two-photon absorption theoretically in 1931 (decades before lasers existed to observe it). The unit is small because the underlying cross-section values genuinely are — two-photon absorption is a weak effect on a per-molecule basis compared to ordinary single-photon absorption, so the GM unit was chosen to keep everyday reported values (roughly 1-1000 GM for common fluorophores) in a convenient range.
Why does tighter focusing (smaller FWHM) increase the excitation probability?
Because a smaller focal spot concentrates the same laser power into a smaller area, raising the peak intensity (I = 2P / (pi w^2)) — and since excitation probability depends on the square of photon flux, which itself depends on intensity, even a modest reduction in spot size produces an outsized increase in excitation probability. This intensity-squared sensitivity to focusing is exactly why two-photon microscopy needs a genuinely tight, diffraction-limited focus to work efficiently.
Does shorter pulse duration always help, for the same average power?
Yes, when peak power (not average power) is what's held fixed relative to pulse duration — this calculator's formula scales excitation linearly with tau, but a shorter pulse at the same average power actually delivers higher peak power (since the same energy is packed into less time), which raises intensity and therefore excitation probability further through the squared photon-flux term. This is exactly why two-photon microscopy uses femtosecond-pulsed lasers rather than continuous-wave ones: concentrating light in time, not just in space, is what makes the nonlinear excitation efficient enough to use.
Why does this need a wavelength input if two-photon excitation uses two identical photons?
Because photon flux itself depends on wavelength — a fixed laser power delivers more individual photons per second at a longer wavelength (since each individual photon carries less energy), which changes the photon flux term the excitation formula is built on. Wavelength is also what determines whether a given fluorophore can be two-photon excited at all, since it must roughly correspond to about twice the energy of that fluorophore's normal single-photon absorption wavelength.