Two-photon absorption (TPA) is a third-order nonlinear optical process that involves a simultaneous interaction of a molecule with two photons.112,113 Typically, when TPA occurs, a single photon is nonresonant with any electronic transition in the molecule but the sum of the two is resonant with some available transition. Although the nonresonant condition for a single photon is not mandatory, it is difficult to disentangle
one- and two-photon transitions in this case. The word instantaneous is crucial in the definition above, as it allows to differentiate TPA from excited-state absorption (ESA) that involves sequential absorption of the photons. TPA scales with the square of light intensity in contrast to the linear dependence for one-photon absorption (OPA). Therefore, it occurs at high instantaneous photon densities such as those available in focused femtosecond beams.
The attenuation of a beam of light with intensity 𝐼 over distance 𝑧 in a medium due to two-photon absorption is given by25
𝜕𝐼
𝜕𝑧= −𝑁𝛼/𝐼/= −𝑁𝜎(/)𝐹𝐼 (1.15) where 𝐹 = 𝐼/ℎ𝜈 is the photon flux, 𝑁 is the number of molecules per unit volume, and 𝛼/ is the molecular coefficient for TPA while 𝜎(/) is the molecular TPA cross-section. For a plane-polarized light, the 𝜎(/) value at the maximum of a Lorentzian-shaped band corresponding to the transition between the ground state 𝑔 and a final state 𝑓 is given by1
𝜎dAe(/) =2𝜋ℎ𝜈/(𝑛/+ 2)f 3f𝜀g/𝑛/𝑐/
1
Γ 𝑆$S (1.16)
where Γ is the half-width at half-maximum of the band, and 𝑆$S is1,114 𝑆$S=1 where 𝜇mn is the amplitude of the transition dipole moment induced by the electric field of the wave whose frequency matches the energy gap between any 𝑘 and 𝑙 states, and ∆𝜇S$ is the change in the permanent dipole moment in the 𝑓 state relative to the 𝑔 state. 𝐸S" is the energy gap between the ground state and intermediate state 𝑖. The appearance of the intermediate state 𝑖 in this equation is very important and is explained below.
Two different cases, corresponding to centrosymmetric quadrupolar and non-centrosymmetric dipolar molecules, are considered now (Figure 1.5). For centrosymmetric molecules, the first term in brackets (so-called dipolar term) in Eq. (1.17) disappears since the permanent dipole
moments in both states 𝑔 and 𝑓 equal zero. The TPA cross-section can be approximated by (𝐶 is a constant):
𝜎dAe(/) ≈ 𝐶 𝜇S"/ 𝜇"$/
(𝐸S"/ℎ𝜈 − 1)/Γ (1.18)
Both ground and final states are symmetric with respect to the center of inversion, whereas the intermediate state is antisymmetric. The transitions are one-photon electric-dipole allowed for both 𝑔→𝑖 and 𝑖→𝑓 transitions.
For TPA, the frequency 𝜈 is out of resonance with both these transitions, but it creates a virtual state that is a superposition of |𝑔 and |𝑖 , in which the induced polarization is detuned from that in the intermediate state |𝑖 by a frequency difference that corresponds to the energy ∆ = 𝐸S"− ℎ𝜈.
This virtual state is, of course, a nonstationary state of the system that exists only while the molecule experiences the field of the first photon, but the transient presence of |𝑖 with ungerade parity in the superposition allows the second photon of frequency 𝜈 to induce an electric dipole transition to the final gerade state |𝑓 .25,115 The 𝑔→𝑓 transition is thus allowed in TPA and forbidden in OPA. This reversal of the selection rules in OPA and TPA is general for all centrosymmetric molecules.
It is worth highlighting that this ‘photon’ picture is chosen here because it provides the most intuitive explanation of why the parity of the initial and final states is identical for TPA, whereas they should be of opposite parity for OPA. Rigorously speaking, it is suboptimal to discuss the nonlinear optical phenomena in terms of photons, but rather the sequential interactions between the matter and electric fields should be used.
In dipolar non-centrosymmetric molecules, the 𝑔→𝑓 transition is OPA-allowed and the dipolar term in Eq. (1.17) is not zero. In this case,
|𝑓 plays the role of |𝑖 and the transition is present in both linear and two-photon absorption. The second term in Eq. (1.17) (so called two-two-photon term) makes a smaller contribution in dipolar chromophores because it relates to higher states (|𝑖 lies above |𝑓 and ∆ > ℎ𝜈). If, however, upper and not the lowest excited states are considered so that there is some state
|𝑖 below |𝑓 , then ∆ < ℎ𝜈, and assuming ∆𝜇S$ = 𝜇S$, which is valid for
chromophores with a substantial charge transfer upon photoexcitation, the dipolar term in Eq. (1.17) is intrinsically smaller than the two-photon term for centrosymmetric systems. This is one of the reasons why dipolar molecular architectures are less efficient two-photon absorbers than their multipolar analogues. The magnitudes 𝜇S"/ and 𝜇S$/ are proportional to the corresponding one-photon oscillator strengths and can be computed from the linear absorption spectrum. However, 𝜇"$/ is rarely determined experimentally and therefore, the approach toward the synthesis of efficient two-photon absorbers is still largely empirical.25 Addition of the donors and acceptor groups to conjugated systems increases the displacement of charge and results in enhanced transition dipole moments.
Multichromophoric and multipolar assemblies also often lead to better efficiency of the TPA process.17,22
Figure 1.5 | Energy level diagrams for the essential states in centrosymmetric quadrupolar and non-centrosymmetric dipolar chromophores. The diagram is general to the lowest TPA transitions in any centrosymmetric or non-centrosymmetric molecule. Adapted from Ref. 25.