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Two-photon absorption (TPA) is the absorption of two photons of identical or different frequencies in order to excite a molecule from one state (usually the ground state) to a higher energy, most commonly an excited electronic state. The energy difference between the involved lower and upper states of the molecule is equal to the sum of the photon energies of the two photons absorbed. Two-photon absorption is a third-order process, typically several orders of magnitude weaker than linear absorption at low light intensities. It differs from linear absorption in that the optical transition rate due to TPA depends on the square of the light intensity, thus it is a nonlinear optical process, and can dominate over linear absorption at high intensities. Two-photon absorption (TPA) is the absorption of two photons of identical or different frequencies in order to excite a molecule from one state (usually the ground state) to a higher energy, most commonly an excited electronic state. The energy difference between the involved lower and upper states of the molecule is equal to the sum of the photon energies of the two photons absorbed. Two-photon absorption is a third-order process, typically several orders of magnitude weaker than linear absorption at low light intensities. It differs from linear absorption in that the optical transition rate due to TPA depends on the square of the light intensity, thus it is a nonlinear optical process, and can dominate over linear absorption at high intensities. Two-photon absorption can lead to two-photon-excited fluorescence where the excited state produced by TPA decays by spontaneous emission to a lower energy state. The phenomenon was originally predicted by Maria Goeppert-Mayer in 1931 in her doctoral dissertation. Thirty years later, the invention of the laser permitted the first experimental verification of TPA when two-photon-excited fluorescence was detected in a europium-doped crystal. Soon afterwards, the effect was observed in cesium vapor and then in CdS, a semiconductor. TPA is a nonlinear optical process. In particular, the imaginary part of the third-order nonlinear susceptibility is related to the extent of TPA in a given molecule. The selection rules for TPA are therefore different from one-photon absorption (OPA), which is dependent on the first-order susceptibility. The relationship between the selection rules for one-and two-photon absorption is analogous to those of Raman and IR spectroscopies. For example, in a centrosymmetric molecule, one- and two-photon allowed transitions are mutually exclusive, an optical transition allowed in one of the spectroscopies is forbidden in the other. In quantum mechanical terms, this difference results from the fact that the quantum states of such molecules have either + or - inversion symmetry, usually labelled by g (for +) and u (for -). One photon transitions are only allowed between states that differ in the inversion symmetry, i.e. g <-> u, while two photon transitions are only allowed between states that have the same inversion symmetry, i.e. g <->g and u <-> u. The relation between the number of photons - or, equivalently, order of the electronic transitions - involved in a TPA process (two) and the order of the corresponding nonlinear susceptibility (third) may be understood using the optical theorem. This theorem relates the imaginary part of an all-optical process of a given perturbation order m {displaystyle m} with a process involving charge carriers with half the perturbation order, i.e. m / 2 {displaystyle m/2} . To apply this theorem it is important to consider that the order in perturbation theory to calculate the probability amplitude of an all-optical χ ( n ) {displaystyle chi ^{(n)}} process is m = n + 1 {displaystyle m=n+1} . Since in the case of TPA there are electronic transitions of the second order involved ( m / 2 = 2 {displaystyle m/2=2} ), it results from the optical theorem that the order of the nonlinear susceptibility is n = m − 1 = 3 {displaystyle n=m-1=3} , i.e. it is a χ ( 3 ) {displaystyle chi ^{(3)}} process. In the next paragraph resonant two photon absorption via separate one-photon transitions is mentioned, where the absorption alone is a first order process and any fluorescence from the final state of the second transition will be of second order; this means it will rise as the square of the incoming intensity. The virtual state argument is quite orthogonal to the anharmonic oscillator argument. It states for example that in a semiconductor, absorption at high energies is impossible if two photons cannot bridge the band gap. So, many materials can be used for the Kerr effect that do not show any absorption and thus have a high damage threshold. Two-photon absorption can be measured by several techniques. Two of them are two-photon excited fluorescence (TPEF) and nonlinear transmission (NLT). Pulsed lasers are most often used because TPA is a third-order nonlinear optical process, and therefore is most efficient at very high intensities. Phenomenologically, this can be thought of as the third term in a conventional anharmonic oscillator model for depicting vibrational behavior of molecules. Another view is to think of light as photons. In nonresonant TPA two photons combine to bridge an energy gap larger than the energies of each photon individually. If there were an intermediate state in the gap, this could happen via two separate one-photon transitions in a process described as 'resonant TPA', 'sequential TPA', or '1+1 absorption'. In nonresonant TPA the transition occurs without the presence of the intermediate state. This can be viewed as being due to a 'virtual' state created by the interaction of the photons with the molecule. The 'nonlinear' in the description of this process means that the strength of the interaction increases faster than linearly with the electric field of the light. In fact, under ideal conditions the rate of TPA is proportional to the square of the field intensity. This dependence can be derived quantum mechanically, but is intuitively obvious when one considers that it requires two photons to coincide in time and space. This requirement for high light intensity means that lasers are required to study TPA phenomena. Further, in order to understand the TPA spectrum, monochromatic light is also desired in order to measure the TPA cross section at different wavelengths. Hence, tunable pulsed lasers (such as frequency-doubled Nd:YAG-pumped OPOs and OPAs) are the choice of excitation. The Beer's law for one photon absorption:

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