Two Photon Absorption Calculator

Two-photon absorption promotes a molecule to an excited state by taking up two photons at essentially the same instant, each supplying about half the transition energy. The intermediate is a virtual state with no real population and a lifetime set by the uncertainty principle at roughly 10 to the minus 15 seconds, so the second photon must arrive within that window. This makes the transition rate proportional to the square of the photon flux rather than to the first power, and the useful consequence is that absorption is confined to the small volume where a focused beam reaches peak intensity. The molecular cross section sigma-2 is quoted in Goeppert-Mayer units, where 1 GM equals 10 to the minus 50 cm to the fourth times seconds per photon per molecule. The macroscopic response is the TPA coefficient beta, obtained by multiplying sigma-2 by the number density N in molecules per cubic centimetre, and for a beam of intensity I traversing path length L the transmission of the simple two-photon model is T = 1/(1 + beta I L). Typical values span a wide range: common fluorophores such as fluorescein and rhodamine sit between about 10 and 300 GM in the 700 to 900 nm window, engineered donor-acceptor and quadrupolar chromophores reach 1,000 to over 10,000 GM, and semiconductor materials are described by bulk beta values in cm per GW instead. Practical work is dominated by the intensity requirement and by measurement artefacts. Continuous-wave sources are hopeless; the field relies on mode-locked titanium-sapphire lasers delivering roughly 100 femtosecond pulses at 80 MHz, which reach focal peak intensities well above 10 to the 11th W per square centimetre while keeping average power low enough to avoid thermal damage. Because rate scales with the square of intensity, halving pulse duration at fixed average power doubles the excitation yield, so pulse dispersion in the optics must be pre-compensated. Reported cross sections vary widely between laboratories: measurements by the open-aperture Z-scan method give beta directly but are prone to contamination from excited-state absorption, thermal lensing, and self-focusing, while two-photon excited fluorescence methods are relatively robust but require an accurate reference standard and a known fluorescence quantum yield. The simple transmission expression used here also assumes a uniform beam, negligible linear absorption at the excitation wavelength, and no saturation or photobleaching, none of which strictly hold in a tight focus.

The process was predicted before any light source could produce it. Maria Goeppert-Mayer worked out the theory of two-photon processes in her 1931 doctoral dissertation at Goettingen under Max Born, publishing the second-order perturbation treatment in Annalen der Physik and deriving the cross section that now carries her name as a unit. She later shared the 1963 Nobel Prize in Physics for unrelated work on the nuclear shell model. Her prediction sat untested for three decades because ordinary lamps cannot supply anything like the required photon flux. Theodore Maiman demonstrated the ruby laser in 1960, and confirmation followed almost immediately: Wolfgang Kaiser and C. G. B. Garrett reported two-photon excited blue fluorescence from europium-doped calcium fluoride in Physical Review Letters in 1961, the first observation of the effect. Through the 1960s and 1970s two-photon spectroscopy became a tool for reaching states forbidden by one-photon selection rules, since the parity requirements differ, giving access to gerade excited states in centrosymmetric molecules. The application that transformed the field came in 1990, when Winfried Denk, James Strickler, and Watt Webb at Cornell University published two-photon laser scanning fluorescence microscopy in Science, using femtosecond near-infrared pulses to confine excitation to the focal plane, which suppressed out-of-focus photobleaching and allowed imaging hundreds of micrometres into living tissue. Shinjiro Maruo, Osamu Nakamura, and Satoshi Kawata demonstrated two-photon stereolithography in 1997, opening sub-diffraction three-dimensional microfabrication, and a sustained effort through the 1990s and 2000s by groups including those of Seth Marder and Joseph Perry established the structure-property rules for designing high cross-section chromophores.