Cooperative two-photon absorption

The transition matrix element for double-beam cooperative two-photon absorption can be obtained by setting A = B and a = in Eq. (5.13), and discarding the forbidden contributions associated with the third and fourth terms. The vectors R and R now become R i and R 2 leading to a matrix element given by ... 

The simplest case is that of single-beam cooperative two-photon absorption, which is the only process where no phased ave rages arise. The pre-averaged rate, which is suitable for two centers that are rig idly held in a fixed orientation... 

Irradiation with a single beam of laser light at the mean of these two frequencies should thus lead to a synergistic two-photon process, involving both the cooperative and distributive channels, in which both species are simultaneously excited. The narrow bandwidth of any standard laser source should ensure that neither species is independently excited by a conventional single-photon absorption process. Evidence for cooperative two-photon absorption is then provided by detection of the decomposition product CO resulting from the reactions ... 

Terenziani F, Parthasarathy V, Pla-Quintana A, Maishal T, Caminade AM, Majoral JP, Blanchard-Desce M (2009) Cooperative two-photon absorption enhancement by through-space interactions in multichromophoric compounds. Angew Chem Int Ed 48 8691-8694... 

Figure 2. A cooperative mechanism for synergistic two-photon absorption. Molecule A ab orl>s a photon of frequency

As seen above, synergistic two-photon absorption can in principle take place by either or both of the mechanisms, where (i) each laser photon is absorbed by a different molecule (the cooperative mechanism), or (ii) both laser photons are absorbed by a single molecule (the distributi e mechanism). In each case, the energy mismatch for the molecular transitior s is transferred between the molecules by means of a virtual photon that couples with each molecule by the same electric-dipole coupling as the laser photons. The result, however, is a significant difference in the selection rules applying to the two types of processes. 

Figure 5. Typical time-ordered diagrams for single-beam two-photon absorption (a) shows one of the diagrams associated with the cooperative mechanism, and (b) one of the diagrams for the distributive mechanism.

The first two terms in Eq. (5.13) arise from the cooperative mechanism, while the distributive mechanism gives rise to the third and fourth terms. Deriving the general rate for a proximity-induced two-photon absorption process from the square modulus of the result is an elaborate procedure producing sixteen terms, including cross-terms associated with quantum mechanical interference between the cooperat ve and distributive mechanisms. However, in view of the selection rules discussed earlier, it is not generally necessary to perform this calculation since each of the four specific mechanisms for two-photon absorption under consideration can, at most, have only two terms of Eq. (5.13) contributing to the matrix element. 

In terms of polarization analysis, double-beam cooperative and doublebeam distributive two-photon absorption display identical behavior since... 

However, reality is not that simple and different up-conversion processes may exist simultaneously, or their effects can be tentatively reinforced reciprocally. For instance, two-photon absorption and cooperative absorption have been theoretically investigated (Rios Leite and de Araujo 1980). Also SHG and cooperative luminescence have been considered simultaneously, in order to increase SHG by the partial resonance of cooperative luminescence (Bonneville and Auzel 1976, Ovsyankin and Fedorov 1981). 

If an ion possesses two excited states with approximately same energy separation, energy transfer can occm between two ions in the first excited state (Figme 4c). One ion returns uomadiatively to the ground state while the second is promoted to the second excited state, then decays with emission of a photon whose energy is about twice that of incident photons. There are other upconversion processes two-step absorption, cooperative sensitization, cooperative luminescence. Upconversion by energy transfer is the most efficient process. 

By introducing the formalism of virtual photon coupling, the timescale for cooperative absorption, t, can be interpreted in terms of a range of propagation for which the exchanged photon has virtual character. Thus the distance R between two molecules that cooperate in the absorption process must be subject to the condition... 

The former relation, Eq. (4.5), indicates the fact that this cooperative process again has the characteristics of mean-frequency absorption here, however, it is the molecular excitation frequency which equals the mean of the two photon frequencies. 

On casting the rate equations entirely in terms of C, it becomes evident that the result for the single-beam cooperative case contains only terms in and numerical terms, while additional terms linear in C occur in the single-beam distributive case. Hence, the odd-j terms in Eq. (7.8) only contribute to the result when circularly or elliptically polarized incident radiation is employed, and their sign is then dependent on the handedness of that incident radiation. A direct consequence of this is the exhibition of two-photon circular dichroism in the distributive absorption process for pairs of molecules with fixed mutual orientations no such effect can occur under the cooperative mechanism. 

One of the most significant implications of the result is that an absorption spectrum measured with intense white light may be significantly different from the spectrum that would be observed using tunable monochromatic radiation. In particular, there should be a decrease in the apparent width of many lines in any absorption spectrum measured with broadband radiation. This is because, for any sample transition of frequency coq, photons of appreciably off-resonant frequency (oiq + fi) can be cooperatively absorbed and result in the excitation of two separate molecules, provided selection rules permit. In fact the Lorentzian linewidth of the concerted absorption process is readily shown to be approximately 0.64 x the ordinary absorption linewidth, if the probe radiation is assumed to be of nearly constant intensity in the frequency region of interest. Nonetheless, the observed linewidth would not be reduced to quite this extent, because of the additional and invariably stronger response associated with normal single-photon absorption. 

With the advent of lasers we have observed the simultaneous absorption of two photons by one electron. Instances of two adjacent molecules cooperatively absorbing one photon are also known. Such occurrences are exceptions to the more normal one photon/one electron phenomena. 

Recombination is either radiative or non-radiative. The radiative process is accompanied by the emission of a photon, the detection of which is the basis of the luminescence experiment. The radiative transition is the inverse of optical absorption and the two rates are related by detailed balance. Non-radiative recombination is commonly mediated by the emission of phonons, although Auger processes are sometimes important, in which a third carrier is excited high into the band. The thermalization process occurs by the emission of single phonons and is consequently very rapid. Non-radiative electron-hole recombination over a large energy requires the cooperation of several phonons, which suppresses the transition probability. 

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