Two-photon absorption

In atomic physics, two-photon absorption, 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, 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, which is proportional to the square of the light intensity 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 and the order of the corresponding nonlinear susceptibility may be understood using the optical theorem. This theorem relates the imaginary part of an all-optical process of a given perturbation order with a process involving charge carriers with half the perturbation order, i.e.. To apply this theorem it is important to consider that the order in perturbation theory to calculate the probability amplitude of an all-optical process is. Since in the case of two-photon absorption there are electronic transitions of the second order involved, it results from the optical theorem that the order of the nonlinear susceptibility is, i.e. it is a process.
There are two 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, where is the i-th component of the polarization field,, etc. are the j-th, etc. components of the three electric fields involved in a third-order process, and 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 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. Note that this convention of absorption for and emission for 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 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, 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 and u. One photon transitions are only allowed between states that differ in the inversion symmetry, i.e., while two photon transitions are only allowed between states that have the same inversion symmetry, i.e. and.
Below are a series of tables outlining the electric-dipole selection rules for two-photon absorption in a bulk material. is the total angular momentum of the state and is the projection of. For the polarization-specific rules, means light linearly polarized along, means light linearly polarized orthogonal to, and means left- and right-circularly polarized light, respectively.

Degenerate and non-degenerate TPA	Degenerate TPA only
	is forbidden
same parity, i.e.,	If, then is forbidden
integer

The polarization-dependence of the TPA selection rules has distinct effects on TPA spectra in semiconductor quantum wells. Light polarized in the plane of the well can excite transitions from the light-hole or the heavy-hole band. However, light polarized normal to the plane of the QW can only excite transitions from the light-hole band.
This follows directly from the selection rule in the table above. In solid-state physics, the LH and HH bands arise from the two different values the valence electrons can take, with HH having and LH having. In the conduction band, we assume all electrons are in s-like states, with . From the table above, under TM polarization, one of the selection rules is . Thus, TM polarized light cannot excite HH-CB transitions. On the other hand, TE polarized light has no such restriction on. 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, z-scan, self-diffraction or nonlinear transmission. 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

describes the decay in intensity due to one-photon absorption:
where are the distance that light travelled through a sample, is the light intensity after travelling a distance, is the light intensity where the light enters the sample and 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
for two-photon absorption with light intensity as a function of path length or cross section as a function of concentration and the initial light intensity. The absorption coefficient now becomes the TPA coefficient.

Two-photon excited fluorescence

Two-photon excitation of a fluorophore 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 is given by
where and are the fluorescence quantum efficiency of the fluorophore and the fluorescence collection efficiency of the measurement system, respectively. In a particular measurement, is a function of fluorophore concentration, illuminated sample volume, incident light intensity, and two-photon absorption cross-section :
Notice that the 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 , where
Considering the reason for these units, one can see that it results from the product of two areas and a time. The large scaling factor is introduced in order that 2-photon absorption cross-sections of common dyes will have convenient values.