Two-Photon Absorption Calculator

Two-photon absorption (2PA) is a nonlinear optical process where a material absorbs two photons simultaneously to make one electronic transition, instead of the usual single-photon absorption. Because it requires two photons at once, the absorption rate scales with the square of the light intensity, so it only becomes significant at high intensities such as pulsed laser beams.

The intensity of light propagating through a 2PA medium follows dI/dz = -alpha*I - beta*I^2, where alpha is the linear absorption coefficient and beta is the two-photon absorption coefficient. When linear absorption is negligible, this integrates to the closed form I(L) = I0 / (1 + beta*I0*L), giving transmittance T = 1 / (1 + beta*I0*L).

For low nonlinear absorption, where beta*I0*L is much smaller than 1, the transmittance change simplifies to dT/T ~= -beta*I0*L. This is the standard working approximation used in early-stage Z-scan analysis (Sheik-Bahae et al., 1990) and is accurate to within a few percent when beta*I0*L is below about 0.1.

The simplified formula dT/T ~= -beta*I0*L is the first-order Taylor expansion of the exact result T - 1 = -q0 / (1 + q0), valid only when q0 = beta*I0*L is small. As intensity rises, q0 grows and the denominator (1 + q0) becomes significant, so the exact expression drops less steeply than the linear approximation predicts.

Beta is most commonly reported in cm/GW in the nonlinear optics literature, paired with intensity in GW/cm2 and path length in cm, which keeps beta x I0 x L dimensionless. Some papers use SI units of m/W instead; always confirm the units before combining values from different sources.

The most common technique is open-aperture Z-scan, where a sample is moved through the focus of a laser beam and the transmitted power is recorded as a function of position. Because intensity peaks at the focus, the normalized transmittance dips there in proportion to beta, and fitting the dip to the closed-form 2PA transmittance curve extracts beta.

One-photon (linear) absorption removes energy proportional to intensity, following Beer-Lambert's law, and does not depend on how intense the beam is per pulse. Two-photon absorption requires two photons to arrive together, so its absorption rate scales with intensity squared, making it negligible at low power but significant under tightly focused pulsed lasers.

Materials with a large beta are used in optical power limiting, three-dimensional microfabrication and two-photon lithography, two-photon fluorescence microscopy, and photodynamic therapy, because the quadratic intensity dependence lets these effects be confined tightly to a focal volume in three dimensions.

Yes. Beta depends strongly on the photon energy relative to the material's band gap or transition energies, and it is typically largest when two photons together have just enough energy to bridge an allowed transition. Because of this, 2PA spectra are used to map out otherwise-forbidden or hard-to-reach electronic transitions.

Under the pure-2PA closed-form solution, I(L) = I0 / (1 + beta*I0*L), where I0 is the input intensity, beta is the 2PA coefficient, and L is the path length. Multiplying I0 by the transmittance T = 1 / (1 + beta*I0*L) gives the same result directly.

No, they are opposite effects. Saturable absorption decreases absorption as intensity increases, because absorbing states become depleted, while two-photon absorption increases absorption as intensity increases, because the two-photon transition rate grows with intensity squared. Some materials exhibit both effects at different intensity ranges.