Twophoton absorption (TPA or 2PA) is a nonlinear absorption process whereby two photons are absorbed simultaneously by an atom, ion or molecule and an electron is promoted from a lower energy level to a higher energy level. For example, twophoton absorption can excite an electron from the ground state to an excited state. The total energy of the transition is equal to the sum of the two photon energies.
The equation describing the intensity attenuation of a light beam passing through a material undergoing singlephoton absorption and twophoton absorption is given by
∂I / ∂z = − α I − β I^{2}  (PROP equation) 
where α and β are constants. The constant β for twophoton absorption is related to the thirdorder susceptibility. The absorbed power in a thickness element dz for singlephoton absorption is proportional to light intensity, whereas the absorbed power for twophoton absorption is proportional to the intensity squared.
Twophoton absorption constant β is usually found for a certain (known) concentration of photoactivated material (species). The main assumption is that during interaction of laser beam with the material the concentration of photoactivated species (molecules, ions, QDs etc) remains constant most of the time (e.g., it reaches equilibrium in a fraction of the beam duration). However, for ultrafast processes this assumption is no longer valid and a more detailed model is usually used for numerical calculations. The central parameter of the new model is a twophoton absorption molecular crosssection σ_{TPA} , while the concentration (or, population density) of the species being in ground state N_{0} is is “detached” from the absorption constant as follows:
∂I(z,t) / ∂z = − σ_{TPA} N_{0}(z,t) I^{2}(z,t)  (detailed PROP) 
For simplicity, we left many other important terms out of this example equation – which may include singlephoton, threephoton, or, generally speaking, multiphoton absorption, linear absorption, stimulated emission etc. Each multiphoton absorption term will have a factor of population density of the energy state from which the absorption occurs.The function of concentration N_{0}(z,t) follows the following corresponding rate equation:
∂N_{0} / ∂t = − σ_{TPA} N_{0}(z,t) I^{2}(z,t) /2 h ω_{0} + k_{10} (NN_{0})  (RATE equation) 
where N is the initial concentration of species and k_{10} attributes to the rate of relaxation of species from the excited state.
See also: Singlephoton absorption, Threephoton absorption, Relaxation.
SimphoSOFT® supports modeling twophoton (as well as single and threephoton) absorption. Its mathematical model includes a variant of PROP equation, written in the terms of electromagnetic field, coupled with timeresolved radiallydependent RATE equations.
SimphoSOFT® case study of twophoton absorption in a chromophore AF455. SimphoSOFT® case study of twophoton absorption in semiconductor quantum dots CdS SimphoSOFT® case study of fitting twophoton absorption of organic dye G74 using Zscan (Video is available) SimphoSOFT® case study of modeling twophoton scanning microscopy to choose the best laser for the money Video describing how to build energy level diagram for optical limiter, containing single and twophoton absorption transitions, and how to optimize its performance in SimphoSOFT®. 
App Notes 
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