Three-Photon Transitions

TRCIS was first applied to dissociative multiphoton ionization of NO2 at 375.3 nm [129]. This was identified as a three-photon transition to arepulsive surface correlating with NO(C2II) + 0(3P) fragments. The NO(C) was subsequently ionizing by a single photon, yielding NO+ (X1 S+) + e. ... 

Here, Te g is the three-photon transition operator, given in Eq. (3.44) as... 

We recalculated the three-photon absorption coefficient, y, making use of the formulas which relate the third order transition dipole moment, three-photon transition probability, 8 and 7 ... 

MW/cm for resonant one-photon transitions 3 -> 2 and 169 GW/cm for resonant three-photon transitions 1 4. For comparison, Are saturation intensity for... 

The development of multiphoton spectroscopy has followed that of lasers as the available power has increased, so has the number of photons involved in individual transitions. More significantly, it has become apparent that the physics of the interaction between radiation and matter is not the same at high laser powers as under weak illumination, i.e. that there is a qualitative change which sets in at strong laser fields. This is normally expressed by saying that perturbative approximations break down. A more direct (and equally accurate) statement is that new effects are observed, which are not present in conventional spectroscopy, even where the latter is extended to include, say, two- and three-photon transitions. 

Figures 3.11 and 3.12 illustrate the weak-field and field-dressed pictures of photodissociation of the Hg molecular ion. In Fig. 3.11, one-, two-, and three-photon transitions from v = 2 of the X2 +(1si7s) bound state to the 2 (2pcru) continuum are shown. The 1- and 3-photon transitions are electronically u — g allowed, but, in a perturbative picture, the 1-photon transition has a vastly smaller Franck-Condon factor than the 3-photon transition. The 2-photon transition to the 2E+ state is (g u)forbidden, but the 2E+ state acts as a virtual...

The selection rules for parity are <-> for N even and for AT odd. Equation (6.1.30) will be explicitly discussed below for the examples of one-, two-, and three-photon transitions. 

It can be shown that for one-color, three-photon transitions, (Dixon, et al., 1984), the electronic transition operator has the general form... 

The three-photon transition probability is proportional to crj3 as defined by Eq. (6.1.30), which includes terms involving the square modulus of scalar products of molecular transition moments and photon polarization tensors. The complicated looking expression is reduced as follows. 

Equations (6.1.30) and (6.1.61) axe valid when the initial and final states are in Hund s case (a). In the case of orbitally degenerate 5 0 states, if either or both of the initial and final states are intermediate between Hund s cases (a) and (b), interference effects can occur, as illustrated by Gauyacq, et al., (1987) for the NO C2n—X2n three-photon transition. 

For molecular transitions the matrix elements Dim and Dmf are composed of three factors the electronic transition dipole element, the Franck-Condon factor, and the Honl-London factor (Sect. 1.7). Within the two-photon dipole approximation a co) becomes zero if one of these factors is zero. The calculation of linestrengths for two- or three-photon transitions in diatomic molecules can be found in [243, 244]. 

Multiphoton absorption of visible photons may result in ionization of atoms or molecules. At a given laser intensity the ion rate Aion( L) recorded as a function of the laser intensity shows narrow maxima if one-, two-, or three-photon resonances occur. If, for instance, the ionization potential (IP) is smaller than 3hco, resonances in the ionization yield are observed, either when the laser frequency co is in resonance with a two-photon transition between the levels /) and /), i.e. (Ef — Et = 2hco), or when autoionizing Rydberg states can be reached by three-photon transitions [267]. 

J.B. Halpem, H. Zacharias, R. Wallenstein, Rotational line strengths in two- and three-photon transitions in diatomic molecules. J. MoL Spectrosc. 79,1 (1980)... 

The measurements were extended to two-photon and three-photon absorption related to d-g and d-h transitions, respectively, in the n = 13-17 states of sodium. In the case of the three-photon transitions, a microwave frequency was swept while an rf frequency was fixed, and two rf photons and one microwave photon were absorbed. The detection scheme was the same as in Ref. 192. The measurements provided accurate quantum defect values and revealed hydrogenic fine structures for I <2. 

The residue analysis of the CRF yields different types of excited-state quantities such as three-photon transition matrix elements (three-photon absorption) [27], the two-photon matrix elements between excited states (the cross section for second-order transitions), and the excited-state polarizability (dynamic second-order property). 

The three-photon transition amplitude has been shown to be obtainable from the single residue of the appropriate CRF [27], although the technique has been employed only rarely. Lin et al. [247] analyzed for instance, solvent effects (accounted for by PCM) on the three-photon absorption spectrum of a symmetric charge transfer molecule using TD-DFT. 

From the discussed residues, we can determine properties explicitly dependent on different electronic states, for instance aU the transition moments coupling the reference state 0) and other states e) (from linear response residues) and aU the one-photon transition moments coupling other pairs of states (from quadratic response double residues). In addition, from single residues of the second- and third-order response functions we can identify two- and three-photon transition matrix elements. 

Three-photon absorption has been described through the single residue of the cubic response function (Cronstrand et al. 2003). Similarly to two-photon absorption, one can discuss the third-order (left and right) transition moments and strengths (Hattig et al. 1998a) and define the three-photon transition strengths in this manner for nonvariational wave functions. There has been a rather limited number of theoretical studies of three-photon absorption processes and these will therefore not be discussed any further here. 

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