Non-Linear Microscopy - Biophotonics - page 56

Fig. 4.3. Perrin-Jablonsky simplified diagram representing the differences between
confocal and two-photon excitation microscopy. Fluorescence emission does not de-
pend on the way in which excitation is performed

where λ 1P is the wavelength required to induce one-photon excitation. For
practical purposes the wavelength of the two photons is chosen identical, so
that

2 hc
λ 2P .

λ 2P = λ 1 = λ 2 =2 λ 1P and ∆ E g = (4.9)

Considering the two-photon process as non-resonant, we can assume the
existence of a virtual intermediate state; the electron will reside in the virtual
state for a time, τ virt , that can be calculated using the time-energy uncertainty
principle:

10 − 15 10 − 16 s .

∆ E g τ virt h/ 2 → τ virt − (4.10)

The two photons will have to be delayed no more than τ virt to induce
the non-linear interaction. It is clear that for a high flux of photons it is
therefore required to have the two-photon interaction. In TPE the photons
spatial and temporal concentration flux will have a crucial role. The shorter
wavelength required for TPE allows using near IR to excite UV and visible
electronic transitions. The rigorous development of multi-photon theory re-
quires to apply perturbation quantum theory [56]. In particular, by solving
the perturbation expansion of time-dependent Shrodinger equation and by us-
ing a Hamiltonian containing the dipole interaction of the molecule with the
incoming radiation, it is possible to show that the first-order solution corre-
sponds to one-photon interaction, while higher order solution are the n -photon
ones [57]. Such a derivation allows to show that TPE depends on the square
of the intensity delivered on the molecule; however, the same result can be
obtained with classical or semi-quantic considerations [58].