Two-Photon Absorption Calculator

Background.

A material that is perfectly transparent to a weak beam can start absorbing it when the light is concentrated enough. In two-photon absorption, a molecule or crystal takes up two photons at the same instant and their combined energy bridges a gap that neither could cross alone. Silicon is the textbook case: a 1550 nm telecom photon carries about 0.8 eV, well short of silicon's 1.12 eV band gap, yet two of them together have enough, and at high intensity silicon waveguides lose light this way.

Because two photons have to arrive together, the process scales with the square of the intensity rather than in proportion to it. The convenient way to describe it is as an extra absorption coefficient that grows with intensity: α₂ = βI, where β is the material's two-photon coefficient and I the local intensity. Double the intensity and the absorption coefficient doubles, so the power absorbed per unit volume rises fourfold.

That square law is what makes two-photon effects useful. In two-photon microscopy, fluorescence comes almost entirely from the focal spot, where the intensity peaks, giving sharp optical sectioning deep in tissue. Two-photon polymerisation writes three-dimensional structures inside a resin the same way, and optical limiters exploit the effect to clamp dangerously intense pulses while passing weak light.

The page evaluates the local value of βI from a coefficient and an intensity you supply. It does not follow the beam as it is absorbed, so it does not give transmission through a thick sample directly, and β itself is only meaningful for the wavelength, pulse duration and material it was measured at. Published values are often quoted in cm/GW, so convert before entering: 1 cm/GW equals 1 × 10⁻¹¹ m/W.

What is two-photon absorption calculator?

The two-photon absorption contribution, α₂, is the part of a material's absorption coefficient that comes from simultaneous absorption of two photons. In the propagation equation dI/dz = −αI − βI², the second term can be written as α₂I with α₂ = βI. It is measured in reciprocal metres, like an ordinary linear absorption coefficient, but unlike one it depends on how intense the light is at that point.

How to use this calculator.

1. 01Enter the two-photon coefficient β in metres per watt. Literature values are usually given in cm/GW; multiply by 1 × 10⁻¹¹ to convert, so 0.5 cm/GW becomes 5 × 10⁻¹² m/W. Use a value measured at your wavelength.
2. 02Enter the optical intensity in watts per square metre. For a pulsed laser use the peak intensity, roughly peak power divided by beam area; for a Gaussian beam the on-axis peak is 2P ÷ (πw²), with w the beam radius.
3. 03To convert from the common unit, 1 GW/cm² is 1 × 10¹³ W/m².
4. 04Read α₂ in m⁻¹ and compare it with the material's linear absorption coefficient to see which loss dominates at that intensity.

The formula.

α₂ = β × I. β, in m/W, is the two-photon absorption coefficient, a property of the material at a given wavelength; I, in W/m², is the optical intensity at the point of interest. Their product has units of m/W × W/m² = m⁻¹, an absorption coefficient. The full attenuation equation is dI/dz = −(α + βI)I, so α₂ adds to the linear coefficient α. Because α₂ itself rises with I, the absorbed power per unit volume, βI², grows as the square of intensity. The result is local: as the beam is absorbed I falls and α₂ falls with it, which is why thick samples need the integrated solution rather than a single α₂.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Boyd, Nonlinear Optics, 4th ed. (PRINT), nonlinear absorption.
2. [2]RP Photonics Encyclopedia, Two-photon Absorption.
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.