Simultaneous absorption of two photons by a molecule
In atomic physics, two-photon absorption (TPA or 2PA), also called two-photon excitation or non-linear absorption, is the simultaneous absorption of two photons of identical or different frequencies in order to excite an atom or a molecule from one state (usually the ground state), via a virtual energy level, to a higher energy, most commonly an excited electronic state. Absorption of two photons with the same frequency is called degenerate two-photon absorption, while absorption of two photons with different frequencies is called non-degenerate two-photon absorption. The energy difference between the involved lower and upper states is equal or smaller than the sum of the photon energies of the two photons absorbed.
Since TPA depends on the simultaneous absorption of two photons, the probability of two-photon absorption is proportional to the photon dose ("D"), which is proportional to the square of the light intensity "D" &amp;propto; "I"2 thus it is a nonlinear optical process. Two-photon absorption is a third-order process, with absorption cross section typically several orders of magnitude smaller than one-photon absorption cross section.
Two-photon absorption 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 two-photon absorption when two-photon-excited fluorescence was detected in a europium-doped crystal. Soon afterwards, the effect was observed in cesium vapor and then in cadmium sulfide, a semiconductor.
Description.
Two-photon absorption is a nonlinear optical process dependent on the third-order nonlinear susceptibility. The relationship between the number of photons - or, equivalently, order of the electronic transitions - involved in a two-photon absorption process (two, in the case of TPA) and the order of the corresponding nonlinear susceptibility (three, in the case of TPA) may be understood using the optical theorem. This theorem relates the imaginary part of an all-optical process of a given perturbation order formula_1 with a process involving charge carriers with half the perturbation order, i.e. formula_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 formula_3process is formula_4. Since in the case of two-photon absorption there are electronic transitions of the second order involved (formula_5), it results from the optical theorem that the order of the nonlinear susceptibility is formula_6, i.e. it is a formula_7process.
There are two (quite orthogonal) models that can be used to understand TPA, namely classical optics and quantum mechanics. In the classical picture, third-order optical process are described by the equation formula_8, where formula_9 is the "i"-th component of the polarization field, formula_10, etc. are the "j"-th, etc. components of the three electric fields involved in a third-order process, and formula_11 is the fourth-rank susceptibility tensor. The tilde over each of these values denotes that they are, in general, complex. TPA can happen when the imaginary part of the relevant formula_12 component is positive. When this value is negative, the opposite process, two-photon emission, can occur. This follows from the same physics that describes single-photon loss and gain in a medium using the first-order equation formula_13. Note that this convention of absorption for formula_14 and emission for formula_15 is the one commonly followed in physics; in engineering, the opposite convention is often used.
In the quantum mechanical model, we think of light as photons. In non-resonant two-photon absorption, neither photon is at resonance with the system energy gap, and two photons combine to bridge the energy gap larger than the energies of each photon individually. If there were an intermediate electronic 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" where the absorption alone is a first order process and the generated fluorescence will rise as the square of the incoming intensity. In non-resonant two-photon absorption 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 two-photon absorption 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 two-photon absorption phenomena. Further, in order to understand the two-photon absorption spectrum, monochromatic light is also desired in order to measure the two-photon absorption cross section at different wavelengths. Hence, tunable pulsed lasers (such as frequency-doubled Nd:YAG-pumped optical parametric oscillators and optical parametric amplifiers) are the choice of excitation.
In a semiconductor, TPA 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 one- or two-photon absorption and thus have a high damage threshold.
Selection Rules.
The selection rules for two-photon absorption are 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. However, for non-centrosymmetric molecules there is no formal mutual exclusion between the selection rules for one-photon absorption and two-photon absorption. 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. formula_16, while two photon transitions are only allowed between states that have the same inversion symmetry, i.e. formula_17 and formula_18.
Below are a series of tables outlining the electric-dipole selection rules for two-photon absorption in a bulk material. formula_19 is the total angular momentum of the state and formula_20 is the formula_21 projection of formula_19. For the polarization-specific rules, formula_23 means light linearly polarized along formula_21, formula_25 means light linearly polarized orthogonal to formula_21, and formula_27 means left- and right-circularly polarized light, respectively. 
The polarization-dependence of the TPA selection rules has distinct effects on TPA spectra in semiconductor quantum wells (QWs). Light polarized in the plane of the well (i.e., TE-polarized) can excite transitions from the light-hole (LH) or the heavy-hole (HH) band. However, light polarized normal to the plane of the QW (i.e., TM-polarized) can only excite transitions from the light-hole band.
This follows directly from the formula_28 selection rule in the table above. In solid-state physics, the LH and HH bands arise from the two different formula_29 values the valence electrons can take, with HH having formula_30 and LH having formula_31. In the conduction band (CB), we assume all electrons are in "s"-like states, with formula_32 (and therefore, with formula_31). From the table above, under TM polarization ("π"-"π" polarization in the table), one of the selection rules is formula_34 (formula_35 in the table). Thus, TM polarized light cannot excite HH-CB transitions. On the other hand, TE polarized light ("σ"-"σ" in the notation of the table above) has no such restriction on formula_36. Thus, both HH-CB and LH-CB transitions can be cause by TE-polarized light. 
Measurements.
Two-photon absorption can be measured by several techniques. Some of them are two-photon excited fluorescence (TPEF), z-scan, self-diffraction or nonlinear transmission (NLT). Pulsed lasers are most often used because two-photon absorption is a third-order nonlinear optical process, and therefore is most efficient at very high intensities.
Absorption rate.
Beer's law describes the decay in intensity due to one-photon absorption:
formula_37
where formula_38 are the distance that light travelled through a sample, formula_39 is the light intensity after travelling a distance formula_38, formula_41 is the light intensity where the light enters the sample and formula_42 is the one-photon absorption coefficient of the sample. In two-photon absorption, for an incident plane wave of radiation, the light intensity versus distance changes to
formula_43
for two-photon absorption with light intensity as a function of path length or cross section formula_44 as a function of concentration formula_45 and the initial light intensity formula_46. The absorption coefficient formula_47 now becomes the TPA coefficient formula_48. (Note that there is some confusion over the term formula_48 in nonlinear optics, since it is sometimes used to describe the second-order polarizability, and occasionally for the molecular two-photon cross-section. More often however, it is used to describe the bulk 2-photon optical density of a sample. The letter formula_50 or formula_51 is more often used to denote the molecular two-photon cross-section.)
Two-photon excited fluorescence.
Two-photon excitation of a fluorophore (a fluorescent molecule) leads to two-photon-excited fluorescence where the excited state produced by two-photon absorption decays by spontaneous emission of a photon to a lower energy state.
Relation between the two-photon excited fluorescence and the total number of absorbed photons per unit time formula_52 is given by
formula_53
where formula_54 and formula_55 are the fluorescence quantum efficiency of the fluorophore and the fluorescence collection efficiency of the measurement system, respectively. In a particular measurement, formula_56 is a function of fluorophore concentration formula_57, illuminated sample volume formula_58, incident light intensity formula_59, and two-photon absorption cross-section formula_50:
formula_61
Notice that the formula_56 is proportional to the square of the incident light as expected for two-photon absorption.
Units of cross-section.
The molecular two-photon absorption cross-section is usually quoted in the units of Goeppert-Mayer (GM) (after its discoverer, Physics Nobel laureate Maria Goeppert-Mayer), where 
 1 GM = 10−50 cm4 s photon−1.
Considering the reason for these units, one can see that it results from the product of two areas (one for each photon, each in cm2) and a time (within which the two photons must arrive to be able to act together). The large scaling factor is introduced in order that 2-photon absorption cross-sections of common dyes will have convenient values.
Development of the field and potential applications.
Until the early 1980s, two-photon absorption was used as a spectroscopic tool. Scientists compared the one-photon absorption and two-photon absorption spectra of different organic molecules and obtained several fundamental structure property relationships. However, in late 1980s, applications started to be developed. Peter Rentzepis suggested applications in 3D optical data storage. Watt Webb suggested microscopy and imaging. Other applications such as 3D microfabrication, optical logic, autocorrelation, pulse reshaping and optical power limiting were also demonstrated.
3D imaging of semiconductors.
It was demonstrated that by using 2-photon absorption charge carriers can be generated spatially confined in a semiconductor device. This can be used to investigate the charge transport properties of such device.
Microfabrication and lithography.
In 1992, with the use of higher laser powers (35 mW) and more sensitive resins/resists, two-photon absorption found its way into lithography. One of the most distinguishing features of two-photon absorption is that the rate of absorption of light by a molecule depends on the square of the light's intensity. This is different from one-photon absorption, where the rate of absorption is linear with respect to input intensity. As a result of this dependence, if material is cut with a high power laser beam, the rate of material removal decreases very sharply from the center of the beam to its periphery. Because of this, the "pit" created is sharper and better resolved than if the same size pit were created using normal absorption.
3D photopolymerization.
In 1997, Maruo "et al." developed the first application of two-photon absorption in 3D microfabrication. In 3D microfabrication, a block of gel containing monomers and a 2-photon active photoinitiator is prepared as a raw material. Application of a focused laser to the block results in polymerization only at the focal spot of the laser, where the intensity of the absorbed light is highest. The shape of an object can therefore be traced out by the laser, and then the excess gel can be washed away to leave the traced solid. Photopolymerization for 3D microfabrication is used in a wide variety of applications, including microoptics, microfluids, biomedical implants, 3D scaffolds for cell cultures and tissue engineering.
Imaging.
The human body is not transparent to visible wavelengths. Hence, one photon imaging using fluorescent dyes is not very efficient. If the same dye had good two-photon absorption, then the corresponding excitation would occur at approximately two times the wavelength at which one-photon